Degree-One Rational Cherednik Algebras for the Symmetric Group
Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized families of Drinfeld orbifold algebras for symmetric groups ac...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
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Інститут математики НАН України
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| Цитувати: | Degree-One Rational Cherednik Algebras for the Symmetric Group. Briana Foster-Greenwood and Cathy Kriloff. SIGMA 17 (2021), 039, 35 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859531259938603008 |
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| author | Foster-Greenwood, Briana Kriloff, Cathy |
| author_facet | Foster-Greenwood, Briana Kriloff, Cathy |
| citation_txt | Degree-One Rational Cherednik Algebras for the Symmetric Group. Briana Foster-Greenwood and Cathy Kriloff. SIGMA 17 (2021), 039, 35 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized families of Drinfeld orbifold algebras for symmetric groups acting on doubled representations that generalize rational Cherednik algebras by deforming in degree one. We characterize rich families of maps recording commutator relations with their linear parts supported only on and only off the identity when the symmetric group acts on the natural permutation representation plus its dual. This produces degree-one versions of 𝖌𝔩ₙ-type rational Cherednik algebras. When the symmetric group acts on the standard irreducible reflection representation plus its dual, there are no degree-one Lie orbifold algebra maps, but there is a three-parameter family of Drinfeld orbifold algebras arising from maps supported only off the identity. These provide degree-one generalizations of the 𝔰𝔩ₙ-type rational Cherednik algebras 𝐻₀ ̦c.
|
| first_indexed | 2026-03-13T07:33:23Z |
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 039, 35 pages
Degree-One Rational Cherednik Algebras
for the Symmetric Group
Briana FOSTER-GREENWOOD a and Cathy KRILOFF b
a) Department of Mathematics and Statistics, California State Polytechnic University,
Pomona, California 91768, USA
E-mail: brianaf@cpp.edu
b) Department of Mathematics and Statistics, Idaho State University,
Pocatello, Idaho 83209, USA
E-mail: cathykriloff@isu.edu
Received August 07, 2020, in final form April 02, 2021; Published online April 19, 2021
https://doi.org/10.3842/SIGMA.2021.039
Abstract. Drinfeld orbifold algebras deform skew group algebras in polynomial degree
at most one and hence encompass graded Hecke algebras, and in particular symplectic
reflection algebras and rational Cherednik algebras. We introduce parametrized families
of Drinfeld orbifold algebras for symmetric groups acting on doubled representations that
generalize rational Cherednik algebras by deforming in degree one. We characterize rich
families of maps recording commutator relations with their linear parts supported only on
and only off the identity when the symmetric group acts on the natural permutation repre-
sentation plus its dual. This produces degree-one versions of gln-type rational Cherednik al-
gebras. When the symmetric group acts on the standard irreducible reflection representation
plus its dual there are no degree-one Lie orbifold algebra maps, but there is a three-parameter
family of Drinfeld orbifold algebras arising from maps supported only off the identity. These
provide degree-one generalizations of the sln-type rational Cherednik algebras H0,c.
Key words: rational Cherednik algebra; skew group algebra; deformations; Drinfeld orbifold
algebra; Hochschild cohomology; Poincaré–Birkhoff–Witt conditions; symmetric group
2020 Mathematics Subject Classification: 16S80; 16E40; 16S35; 20B30
1 Introduction
Skew group or smash-product algebras S(V )#G twist the symmetric algebra S(V ) of a finite-
dimensional vector space V together with the action of the group algebra CG of a finite group G
acting linearly on V . The center is the invariant polynomial ring S(V )G and there is a natural
grading by polynomial degree, with elements in V of degree one and elements in CG of degree
zero.
Utilizing parameter maps that originate as Hochschild 2-cocycles to explore formal deforma-
tions of S(V )#G has proven useful because although the resulting algebras are noncommutative
they give rise to deformations of S(V )G (by examining centers), yet are easily described as quo-
tient algebras. Both the polynomial degree and the support, i.e., which group elements appear
in the nonzero image, are helpful descriptors for the parameter maps and hence the relations
for the quotient algebras.
Degree-zero deformations of skew group algebras involve parameter maps that identify com-
mutators of elements in V with certain elements of the group algebra. Several important families
of these are of broad interest in noncommutative geometry, combinatorics, and representation
theory and are the subject of an already extensive literature (see [12] and [13] and further
references therein). By comparison, finding elements of degree one with which to identify com-
mailto:brianaf@cpp.edu
mailto:cathykriloff@isu.edu
https://doi.org/10.3842/SIGMA.2021.039
2 B. Foster-Greenwood and C. Kriloff
mutators of elements in V requires a more intricate analysis of which cocycles pass obstructions
in cohomology in order to determine if the resulting deformations satisfy PBW properties [8, 18].
As a result, degree-one deformations are not as well understood or as often studied, yet could
also be significant in giving insight into deformations of the invariant algebra S(V )G and in con-
nection with singularities of orbifolds.
Degree-zero deformations of skew group algebras are called Drinfeld graded Hecke algebras
in recognition of their origins in [4] (see also [15]). These include the important special cases
when G acts on a symplectic vector space, and more particularly when G is a complex reflection
group acting by the sum of a reflection representation and its dual (a doubled representation).
The latter leads to the rational Cherednik algebras, first introduced in [3] as rational degenera-
tions of double affine Hecke algebras and later highlighted as an important subfamily of the more
general symplectic reflection algebras introduced in [6]. When built from an action of the sym-
metric group, rational Cherednik algebras model Hamiltonian reduction in quantum mechanics
and are used to show integrability of Calogero–Moser systems [5].
Degree-one deformations of skew group algebras were termed Drinfeld orbifold algebras and
characterized via explicit PBW conditions on parameter maps in [18], building on [1] and [2]. The
conditions are also interpreted in Hochschild cohomology. In [8] we describe the Drinfeld orbifold
algebras for Sn acting on its natural permutation representation, W ∼= Cn, by starting with
candidate 2-cocycles and imposing the PBW conditions from [18]. Here we expand on that class
of examples by considering Sn acting on both its doubled permutation representation W ∗ ⊕W
and the doubled representation h∗ ⊕ h, where W = h⊕ ι is the sum of the (n− 1)-dimensional
irreducible standard and the trivial representations. This not only results in much richer families
of algebras, but also yields degree-one generalizations of rational Cherednik algebras for these
doubled representations.
More specifically, in [8] we describe all Drinfeld orbifold algebras where the linear parts
of the maps recording commutator relations are supported only on or only off the identity in Sn,
and show there are no such maps with linear part supported both on and off the identity.
For the two doubled representations of Sn considered here we describe all degree-one families
of Drinfeld orbifold algebras whose maps have linear part supported only on or only off the
identity (Theorems 7.1 and 7.2). For maps with linear part supported both on and off the
identity we provide a family of examples involving W ∗ ⊕W (Theorem 5.10) and observe there
are no corresponding such maps for the doubled standard representation h∗ ⊕ h (Remark 6.6).
We summarize our main results.
Theorem 1.1. For the symmetric group Sn (n ≥ 3) acting on V ∼= C2n by the doubled permu-
tation representation, there is
(1) a 17-parameter family of Lie orbifold algebras described by 22 homogeneous quadratic equa-
tions, and
(2) a seven-parameter family of Drinfeld orbifold algebras described in terms of parameter maps
with linear part supported only off the identity that are controlled by four homogeneous
quadratic equations in six of the parameters.
These are the only degree-one deformations of the skew group algebra for Sn acting by the doubled
permutation representation whose parameter maps have linear part supported only on or only off
the identity.
See Theorems 4.1 and 5.9 for more details about the maps, Theorems 7.1 and 7.2 for the
resulting quotient algebras, and Table 2 in Section 7.1 for a summary.
Theorem 1.2. For the symmetric group Sn (n ≥ 3) acting on V ∼= C2n−2 by the doubled stan-
dard representation, there are no degree-one Lie orbifold algebras, but there is a three-parameter
Degree-One Rational Cherednik Algebras for the Symmetric Group 3
family of Drinfeld orbifold algebras described by parameter maps with linear part supported only
off the identity.
These are the only degree-one deformations of the skew group algebra for Sn acting by the
doubled standard representation whose parameter maps have linear part supported only on or
only off the identity.
See Theorems 6.3 and 6.4 for details, Theorem 7.3 for the resulting algebras, and Table 3
in Section 7.2 for a summary. The S2 case in Theorems 1.1 and 1.2 can be analyzed in a similar
way but there are some differences in the dimensions of spaces of pre-Drinfeld orbifold algebra
maps and in the explicit PBW conditions.
The algebras in Theorems 1.1 and 1.2 specialize to the well-known rational Cherednik alge-
bras for the symmetric group, described as of gln- and sln-type respectively in [9], and hence
should be of substantial interest. In particular, Theorem 1.2 provides a degree one version
of the sln-type rational Cherednik algebras H0,c. We refer to the algebras as degree-one ratio-
nal Cherednik algebras. Investigating their structure, properties, combinatorics, representation
theory, geometric significance, and potential importance in physics should provide fertile ground
for future research. It would also be natural to explore whether similar algebras exist for other
complex reflection groups.
The paper is organized as follows. After a brief summary of preliminaries in Section 2 that
apply to any finite group acting linearly on Cn, we restrict to the setting of the symmetric
group and the two doubled representations of interest, except as noted in Lemmas 3.1 and 5.1,
Proposition 6.1, and Corollary 6.2. All pre-Drinfeld orbifold algebra maps for Sn acting by the
doubled permutation representation are constructed in Section 3. We analyze when these lift
in Sections 4 and 5, proving Theorems 4.1, 5.9, and 5.10 using computational details treated
earlier in the two sections. In particular, Sections 4.1–4.3 provide explicit equations governing
the parameter maps described in Theorem 4.1 and Section 4.5 provides some related algebraic
varieties that may be of independent interest. Section 6 begins with Proposition 6.1 providing
conditions under which we can combine Drinfeld orbifold algebra maps for subrepresentations
into a map for their direct sum. Corollary 6.2 is then used with the results from Sections 4
and 5 to describe in Theorems 6.3 and 6.4 all Drinfeld orbifold algebra maps for Sn acting by
the doubled standard representation on the subspace h∗ ⊕ h when the linear part is supported
only on or only off the identity. In Section 7 we present as quotients the resulting degree-one
rational Cherednik algebras arising from the maps in Sections 4–6.
2 Preliminaries
Throughout this section, we let G be a finite group acting linearly on a vector space V ∼= Cn.
All tensors will be over C.
2.1 Skew group algebras
Let G be a finite group that acts on a C-algebra R by algebra automorphisms, and write gs
for the result of acting by g ∈ G on s ∈ R. The skew group algebra R#G is the semi-direct
product algebra R o CG with underlying vector space R ⊗ CG and multiplication of simple
tensors defined by
(r ⊗ g)(s⊗ h) = r(gs)⊗ gh
for all r, s ∈ R and g, h ∈ G. The skew group algebra becomes a G-module by letting G act
diagonally on R⊗ CG, with conjugation on the group algebra factor:
g(s⊗ h) = (gs)⊗ (gh) = (gs)⊗ ghg−1.
4 B. Foster-Greenwood and C. Kriloff
In working with elements of skew group algebras, we commonly omit tensor symbols unless the
tensor factors are lengthy expressions.
If G acts linearly on a vector space V ∼= Cn, then G also acts on the tensor algebra T (V )
and symmetric algebra S(V ) by algebra automorphisms. Assign elements of V degree one and
elements of G degree zero to make the skew group algebras T (V )#G and S(V )#G graded
algebras.
2.2 Cochains
A k-cochain is a G-graded linear map µ =
∑
g∈G µgg with components µg :
∧k V → S(V ).
If each µg maps into V , then µ is called a linear cochain, and if each µg maps into C, then µ is
called a constant cochain.
We regard a map µ on
∧k V as a multilinear alternating map on V k and write µ(v1, . . . , vk)
in place of µ(v1 ∧ · · · ∧ vk). Of course, if µ(v1, . . . , vk) = 0, then µ is zero on any permutation
of v1, . . . , vk. Also, if µ is zero on all k-tuples of basis vectors, then µ is zero on any k-tuple
of vectors. We exploit these facts in the computations in Sections 4 and 5.
The support of a cochain µ is the set of group elements for which the component µg is not
the zero map. For X a subset of G, we say a cochain µ is supported only on X if µg = 0 for all g
not in X. Similarly, we say µ is supported only off X if µg = 0 for all g in X. At times, it is
convenient to talk about support in a weaker sense, so we say µ is supported on X if µg 6= 0 for
some g in X and that µ is supported off X if µg 6= 0 for some g not in X. (Hence, it is possible
for a cochain to be simultaneously supported on and off of a set.) The kernel of a cochain µ is
the set of vectors v0 such that µ(v0, v1, . . . , vk−1) = 0 for all v1, . . . , vk−1 ∈ V .
The group G acts on the components of a cochain. Specifically, for a group element h and
component µg, the map hµg is defined by
(
hµg
)
(v1, . . . , vk) = h
(
µg
(
h−1
v1, . . . ,
h−1
vk
))
. In turn,
the group acts on the space of cochains by letting hµ =
∑
g∈G
hµg ⊗ hgh−1. Thus µ is a G-
invariant cochain if and only if hµg = µhgh−1 for all g, h ∈ G.
2.3 Drinfeld orbifold algebras
For a parameter map κ = κL+κC , where κL is a linear 2-cochain and κC is a constant 2-cochain,
the quotient algebra
Hκ = T (V )#G/
〈
vw − wv − κL(v, w)− κC(v, w) | v, w ∈ V
〉
is called a Drinfeld orbifold algebra if the associated graded algebra grHκ is isomorphic to the
skew group algebra S(V )#G. The condition grHκ ∼= S(V )#G is called a Poincaré–Birkhoff–
Witt (PBW) condition, in analogy with the PBW Theorem for universal enveloping algebras.
Further, if Hκ is a Drinfeld orbifold algebra and t is a complex parameter, then
Hκ,t := T (V )#G[t]/
〈
vw − wv − κL(v, w)t− κC(v, w)t2 | v, w ∈ V
〉
is called a Drinfeld orbifold algebra over C[t]. In [18, Theorem 2.1], Shepler and Witherspoon
make an explicit connection between the PBW condition and deformations in the sense of Ger-
stenhaber [10] by showing how to interpret Drinfeld orbifold algebras over C[t] as formal defor-
mations of the skew group algebra S(V )#G. For more on the broader context of formal defor-
mations see [8, Section 4].
2.4 Lie orbifold algebras
The parameter maps of Drinfeld orbifold algebras decompose as κ =
∑
g κgg. When κ is a para-
meter map for a Drinfeld orbifold algebra and the linear part κL = κL1 is supported only on the
Degree-One Rational Cherednik Algebras for the Symmetric Group 5
identity then the map gives rise to a Lie orbifold algebra (see [18, Section 4] and Definition 2.1).
Lie orbifold algebras deform universal enveloping algebras twisted by a group action just as
certain symplectic reflection algebras deform Weyl algebras twisted by a group action.
2.5 Drinfeld orbifold algebra maps
Though the defining PBW condition for a Drinfeld orbifold algebra Hκ involves an isomorphism
of algebras, Shepler and Witherspoon proved an equivalent characterization [18, Theorem 3.1]
in terms of properties of the parameter map κ.
Definition 2.1. Let κ = κL+κC where κL is a linear 2-cochain and κC is a constant 2-cochain,
and let Alt3 denote the alternating group on three elements. Let V g denote the set of vectors
in V that are fixed by group element g. We say κ is a Drinfeld orbifold algebra map if the
following conditions are satisfied for all g ∈ G and v1, v2, v3 ∈ V :
imκLg ⊆ V g, (2.1)
the map κ is G-invariant, (2.2)∑
σ∈Alt3
κLg (vσ(2), vσ(3))(
gvσ(1) − vσ(1)) = 0 in S(V ), (2.3)
∑
σ∈Alt3
∑
xy=g
κLx
(
vσ(1) + yvσ(1), κ
L
y (vσ(2), vσ(3))
)
= 2
∑
σ∈Alt3
κCg (vσ(2), vσ(3))(
gvσ(1) − vσ(1)), (2.4)
∑
σ∈Alt3
∑
xy=g
κCx
(
vσ(1) + yvσ(1), κ
L
y (vσ(2), vσ(3))
)
= 0. (2.5)
In the special case when the linear component κL of a Drinfeld orbifold algebra map is supported
only on the identity, we call κ a Lie orbifold algebra map.
To simplify reference to the expressions appearing in the last three Drinfeld orbifold algebra
map properties, we define operators ψ and φ that convert 2-cochains (such as κL and κC) into
the 3-cochains we see evaluated within the properties.
Definition 2.2. Let µ denote a linear or constant 2-cochain and ν a linear 2-cochain. Define
ψ(µ) =
∑
g∈G ψgg to be the 3-cochain with components ψg :
∧3 V → S(V ) given by
ψg(v1, v2, v3) =
∑
σ∈Alt3
µg(vσ(1), vσ(2))(
gvσ(3) − vσ(3)).
Define φ(µ, ν) =
∑
g∈G φgg to be the 3-cochain with components φg =
∑
xy=g φx,y, where
φx,y :
∧3 V → V ⊕ C is given by
φx,y(v1, v2, v3) =
∑
σ∈Alt3
µx(vσ(1) + yvσ(1), νy(vσ(2), vσ(3))). (2.6)
Thus φ(µ, ν) is G-graded with components φg and also (G×G)-graded with components φx,y.
For the interested reader, we indicate in [8] how the maps ψ and φ relate to coboundary and
bracket operations in Hochschild cohomology of a skew group algebra.
2.6 Drinfeld orbifold algebra maps (condensed definition)
Equipped with the definitions of ψ and φ, the properties of a Drinfeld orbifold map κ = κL+κC
(Definition 2.1) may be expressed succinctly:
6 B. Foster-Greenwood and C. Kriloff
(2.1) imκLg ⊆ V g for each g in G,
(2.2) the map κ is G-invariant,
(2.3) ψ
(
κL
)
= 0,
(2.4) φ
(
κL, κL
)
= 2ψ
(
κC
)
,
(2.5) φ
(
κC , κL
)
= 0.
Note that any G-invariant 2-cochain whose linear part is supported only on the identity
trivially satisfies properties (2.1) and (2.3), so in this case it is enough to analyze conditions
under which properties (2.4) and (2.5) hold (see Theorem 4.1 and Section 4).
Remark 2.3. If Hκ is a Drinfeld orbifold algebra, then κ must satisfy conditions (2.2)–(2.5),
but not necessarily the image constraint (2.1). However, [18, Theorem 7.2(ii)] guarantees there
will exist a Drinfeld orbifold algebra Hκ̃ such that Hκ̃ ∼= Hκ as filtered algebras and κ̃ satisfies
the image constraint im κ̃Lg ⊆ V g for each g in G. Thus, in classifying Drinfeld orbifold algebras,
it suffices to only consider Drinfeld orbifold algebra maps.
Theorem 2.4 ([18, Theorems 3.1 and 7.2(ii)]). A quotient algebra Hκ satisfies the PBW con-
dition grHκ ∼= S(V )#G if and only if there exists a Drinfeld orbifold algebra map κ̃ such that
Hκ ∼= Hκ̃.
2.7 Strategy
As described and utilized in [8], the process of determining the set of all Drinfeld orbifold
algebra maps consists of two phases, and language from cohomology and deformation theory
can be used to describe each phase. First, one finds all pre-Drinfeld orbifold algebra maps,
i.e., all G-invariant linear 2-cochains satisfying the image condition (2.1) and the mixed Jacobi
identity (2.3). To find such maps supported on transpositions (Proposition 3.4) we utilize
a bijection between pre-Drinfeld orbifold algebra maps and a particular set of representatives
of Hochschild cohomology classes (see Lemma 2.5). But to find such maps supported only on the
identity (Proposition 3.2) we present a simpler argument based on Lemma 3.1 analyzing the
eigenvector structure of images dependent on the group action on input vectors. In the second
phase (Sections 4 and 5) we determine for which pre-Drinfeld orbifold algebra maps κL there
exists a compatible G-invariant constant 2-cochain κC such that properties (2.4) and (2.5) hold.
We say κC clears the first obstruction if property (2.4) holds and clears the second obstruction
if property (2.5) holds. If a G-invariant constant 2-cochain κC clears both obstructions, then we
say κL lifts to the Drinfeld orbifold algebra map κ = κL + κC .
2.8 Hochschild cohomology to pre-DOA maps
We briefly recall how Hochschild cohomology can be used in general to find linear and constant
2-cochains κ that are both G-invariant and satisfy property (2.3). For more detailed background
discussion about the connections to deformations and further references see [8].
For an algebra A over C with bimodule M , the Hochschild cohomology of A with coefficients
in M is HH•(A,M) := Ext•A⊗Aop(A,M), which is abbreviated to HH•(A) if M = A. For any
finite group G acting linearly on a vector space V ∼= Cn, and for A = S(V )#G, using results
of Ştefan [19] yields the following, where RG denotes the set of elements in R fixed by every g
in G,
HH•(S(V )#G) ∼= HH•(S(V ), S(V )#G)G ∼= (H•)G.
Degree-One Rational Cherednik Algebras for the Symmetric Group 7
Here H• is the G-graded vector space H• =
⊕
g∈GH
•
g with components
Hp,d
g = Sd(V g)⊗
p−codim(V g)∧
(V g)∗ ⊗
codim(V g)∧ (
(V g)∗
)⊥ ⊗ Cg,
first described independently by Farinati [7] and Ginzburg–Kaledin [11]. Note that H• is tri-
graded by cohomological degree p, homogeneous polynomial degree d, and group element g.
Since the exterior factors of Hp,d
g can be identified with a subspace of
∧p V ∗, and since
Sd(V g)⊗
∧p V ∗⊗Cg ∼= Hom
(∧p V, Sd(V g)g
)
, the space H• may be identified with a subspace
of the cochains introduced earlier in this section. The next lemma records the relationship bet-
ween properties (2.2) and (2.3) of a Drinfeld orbifold algebra map and Hochschild cohomology.
When d = 1, the lemma is a restatement of [18, Theorem 7.2(i) and (ii)]. When d = 0, the
lemma is a restatement of [17, Corollary 8.17(ii)]. It is also possible to give a linear algebraic
proof in the spirit of [16, Lemma 1.8].
Lemma 2.5. For a 2-cochain κ =
∑
g∈G κgg with imκg ⊆ Sd(V g) for each g ∈ G, the following
are equivalent:
(a) The map κ is G-invariant and satisfies the mixed Jacobi identity, i.e., for all v1, v2, v3 ∈ V
[v1, κ(v2, v3)] + [v2, κ(v3, v1)] + [v3, κ(v1, v2)] = 0 in S(V )#G,
where [·, ·] denotes the commutator in S(V )#G.
(b) For all g, h ∈ G and v1, v2, v3 ∈ V :
(i) h(κg(v1, v2)) = κhgh−1
(
hv1,
hv2
)
, and
(ii) κg(v1, v2)(
gv3 − v3) + κg(v2, v3)(
gv1 − v1) + κg(v3, v1)(
gv2 − v2) = 0.
(c) The map κ is an element of
(H2,d)G =
(⊕
g∈G
(
Sd(V g)g ⊗
2−codim(V g)∧
(V g)∗ ⊗
codim(V g)∧ (
(V g)∗
)⊥))G
.
Remark 2.6. Part (b(ii)) of Lemma 2.5 is 2ψ(κ) = 0. Part (c) of Lemma 2.5 shows that κ can
only be supported on elements g with codimV g ∈ {0, 2} since negative exterior powers are zero
and an element g with codimension one acts nontrivially on H2,d
g .
3 Pre-Drinfeld orbifold algebra maps
Except as noted in Lemmas 3.1 and 5.1, Proposition 6.1, and Corollary 6.2, for the rest
of the paper, let G = Sn be the symmetric group with n ≥ 3, let W ∼= Cn denote its nat-
ural permutation representation, and consider the doubled permutation representation of Sn
on V = W ∗⊕W ∼= C2n. Let By = {y1, . . . , yn} be the standard basis forW and Bx = {x1, . . . , xn}
be the corresponding dual basis for W ∗. Then the action of σ ∈ Sn is given by σyi = yσ(i)
and σxi = xσ(i). Recall that W ∗ ∼= h∗ ⊕ ι∗, where Sn acts trivially on the 1-dimensional sub-
space ι∗ of W ∗ spanned by x[n] =
n∑
i=1
xi and by the standard reflection representation on its
(n − 1)-dimensional orthogonal complement h∗, and similarly W ∼= h ⊕ ι. In Remark 4.4 and
Section 6 we also consider the doubled standard representation of Sn on the subspace h∗ ⊕ h
spanned by{
x̄i := xi −
1
n
x[n], ȳi := yi −
1
n
y[n]
∣∣∣∣ 1 ≤ i ≤ n} (3.1)
or by {xi − xj , yi − yj | 1 ≤ i, j ≤ n}.
8 B. Foster-Greenwood and C. Kriloff
In this section we identify all pre-Drinfeld orbifold algebra maps for Sn with n ≥ 3 acting by
the doubled permutation representation on W ∗ ⊕W . That is, we find all linear 2-cochains κL
satisfying the image condition (2.1), the G-invariance condition (2.2), and the mixed Jacobi iden-
tity ψ
(
κL
)
= 0 (2.3). To organize computations we make use of Lemma 2.5 relating Hochschild
cohomology and pre-Drinfeld orbifold algebra maps.
By Remark 2.6, we need only consider group elements whose fixed point space has codimen-
sion zero or two. Thus for Sn acting by the doubled permutation representation we consider
two cases: κL supported only on the identity and κL supported only on the set of transpositions
(which act as reflections on W and bireflections on W ∗ ⊕W ).
3.1 Pre-Drinfeld orbifold algebra maps supported only on the identity
We first prove a lemma that describes all G-invariant maps κ1 :
∧2 V → V ⊕C, where G is any
finite group and V is a permutation representation of G. Since properties (2.1) and (2.3) are
trivially satisfied when g = 1, this will produce pre-Drinfeld orbifold algebra maps supported
only on the identity. Recall that κ1 is G-invariant if and only if
κ1(
gu, gv) = g(κ1(u, v)) (3.2)
for all g in G and u, v ∈ V . The following lemma shows that how G acts on a set of representative
basis vector pairs determines a G-invariant linear cochain.
Lemma 3.1. Suppose G is a finite group acting on a complex vector space V by a permutation
representation. If κL1 is G-invariant, then the following two conditions hold for all g in G and
all basis vector pairs vi and vj.
(i) If g swaps vi and vj, then κL1 (vi, vj) is an eigenvector of g with eigenvalue −1.
(ii) If g fixes vi and vj, then the vector κL1 (vi, vj) is in the fixed space V g.
Suppose every ordered pair v, w of basis vectors can be related to some vi, vj in a set S of repre-
sentative basis vector pairs by v = gvi and w = gvj or v = gvj and w = gvi for some g ∈ G.
If κL1 : S → V satisfies (i) and (ii) for all representative pairs in S, then there is a unique way
to extend κL1 to be G-invariant on
∧2 V .
Proof. Assume κL1 is G-invariant. By (3.2), if g fixes both vi and vj , then κL1 (vi, vj) must be
an element of V g. And if g swaps vi and vj , then due to skew-symmetry, κL1 (vi, vj) must be
a (−1)-eigenvector of g.
Suppose κL1 :
∧2 V → V satisfies (i) and (ii) for a set of representative basis vector pairs.
If v = gvi = hvi and w = gvj = hvj for some representative pair vi, vj and some g, h ∈ G, then
h−1g fixes the basis vector pair vi, vj . Hence by (ii), κL1 (vi, vj) is in V h−1g and gκL1 (vi, vj) =
hκL1 (vi, vj). If instead v = gvi = hvj and w = gvj = hvi, then h−1g swaps vi and vj . Hence by (i),
gκL1 (vi, vj) = hκL1 (vj , vi). These imply that the unique way to extend κL1 to be G-invariant is
well-defined. �
We now apply this to the doubled permutation representation of Sn on W ∗⊕W equipped with
the basis Bx ∪ By, where By = {y1, . . . , yn} is the standard basis for W and Bx = {x1, . . . , xn}
is the corresponding dual basis for W ∗. The following proposition summarizes the definitions
of all G-invariant skew-symmetric bilinear maps, i.e., describes
(
H2,0
1 ⊕H
2,1
1
)G
.
Proposition 3.2. Let Sn (n ≥ 3) act by the doubled permutation representation on V =
W ∗ ⊕W ∼= C2n equipped with basis Bx ∪ By. The Sn-invariant linear and constant 2-cochains
κL1 :
∧2 V → V and κC1 :
∧2 V → C are as given in Definition 3.7 in terms of complex parame-
ters ak, bk for 1 ≤ k ≤ 7 and α, β in C respectively.
Degree-One Rational Cherednik Algebras for the Symmetric Group 9
Proof. Linear cochains. If V ∼= C2n is the doubled permutation representation of Sn and
κL1 :
∧2 V → V is an Sn-invariant map, then the value on any pair of basis vectors in Bx ∪By =
{x1, . . . , xn, y1, . . . , yn} can be obtained by acting by an appropriate permutation on one of the
representative values κL1 (x1, x2), κ
L
1 (y1, y2), κ
L
1 (x1, y1), or κL1 (x1, y2).
Consider κL1 (x1, x2). The permutation σ = (12) swaps x1 and x2, so by Lemma 3.1, κL1 (x1, x2)
must be a (−1)-eigenvector of (12), i.e., a linear combination of the vectors x1− x2 and y1− y2.
Both of these vectors are fixed by the group S{3,...,n} of permutations that fix both x1 and x2.
Thus, for a choice of complex parameters a1 and b1, we let
κL1 (x1, x2) = a1(x1 − x2) + b1(y1 − y2).
A similar argument shows we can let
κL1 (y1, y2) = a2(x1 − x2) + b2(y1 − y2)
for some choice of complex parameters a2 and b2.
Consider κL1 (x1, y1). There are no permutations that will swap x1 and y1. The group of per-
mutations that fix both x1 and y1 is S{2,...,n}, so by Lemma 3.1, κL1 (x1, y1) must be an element
of the subspace
V S{2,...,n} = Span
{
x1, x[n], y1, y[n]
}
.
We define κL1 (x1, y1) to be a linear combination of the basis elements, using complex parameters
a3, a4, b3, and b4 as weights. Orbiting yields the definition
κL1 (xi, yi) = a3xi + a4x[n] + b3yi + b4y[n]
for 1 ≤ i ≤ n.
Consider κL1 (x1, y2). There are no permutations that will swap x1 and y2. The group of per-
mutations that fix both x1 and y2 is S{3,...,n}, so once again by Lemma 3.1, κL1 (x1, y2) must be
an element of the subspace
V S{3,...,n} = Span
{
x1, x2, x[n], y1, y2, y[n]
}
.
We define κL1 (x1, y2) to be a linear combination of the basis elements using complex parameters
a5, a6, a7, b5, b6, and b7 as weights. Orbiting yields the definition
κL1 (xi, yj) = a5xi + a6xj + a7x[n] + b5yi + b6yj + b7y[n]
for 1 ≤ i, j ≤ n with i 6= j.
Constant cochains. By comparison there is only a two-parameter family of G-invariant con-
stant cochains. First, using (3.2), if a constant cochain κC1 :
∧2 V → C is Sn-invariant and some
element g ∈ Sn swaps vi and vj , then κC1 (vi, vj) = 0. Thus κC1 (xi, xj) = κC1 (yi, yj) = 0. Also due
to Sn-invariance, the value of κC1 (xi, yj) only depends on whether i = j or i 6= j, so for α, β ∈ C,
we can let κC1 (xi, yi) = α and κC1 (xi, yj) = β for i 6= j. This shows we have a 2-dimensional
space of Sn-invariant maps κC1 . �
Remark 3.3. It is also possible to confirm the dimensions for the linear and constant invariant
cochains by using the equivalences from Lemma 2.5 and calculating inner products of characters.
If n ≥ 3, then, omitting details, we find for the linear cochains,
dim
(
H2,1
1
)Sn = dim
(
V ⊗
∧2
V ∗
)Sn
=
〈
χι, χV χ∧2 V
〉
=
〈
χV , χ∧2 V
〉
= 14,
and for the constant cochains,
dim
(
H2,0
1
)Sn = dim
(∧2
V ∗
)Sn
= 〈χι, χ∧2 V 〉 = 2,
as expected.
10 B. Foster-Greenwood and C. Kriloff
3.2 Pre-Drinfeld orbifold algebra maps supported only off the identity
The following proposition describes
(
H2,0
g ⊕H2,1
g
)G
where g is a transposition.
Proposition 3.4. Let Sn (n ≥ 3) act by the doubled permutation representation on V = W ∗ ⊕
W ∼= C2n, equipped with the basis Bx∪By. The Sn-invariant linear and constant 2-cochains that
sastify the mixed Jacobi identity and are supported only on transpositions are the maps of the
form given in Definition 3.8.
Proof. We find the centralizer invariants
(
H2,0
g ⊕ H2,1
g
)Z(g)
when g is a transposition. Let
g = (12) and first note that the centralizer of g is Z(g) = 〈(12)〉×S{3,...,n}, the fixed point space
of g is
V g = Span{x1 + x2, x3, . . . , xn, y1 + y2, y3, . . . , yn},
and the orthogonal complement is
(V g)⊥ = Span{x1 − x2, y1 − y2}.
A basis for
∧2((V g)⊥)∗ is the volume form
vol⊥g := (x∗1 − x∗2) ∧ (y∗1 − y∗2).
Note that Z(g) acts trivially on vol⊥g , so(
H2,0
g
)Z(g)
= H2,0
g =
∧2 (
(V g)⊥
)∗ ⊗ Cg = Span
{
vol⊥g ⊗ (12)
}
and (
H2,1
g
)Z(g)
= V Z(g) ⊗
∧2 (
(V g)⊥
)∗ ⊗ Cg
= Span
{
v ⊗ vol⊥g ⊗ (12)
∣∣∣∣ v ∈ {x1 + x2,
n∑
i=3
xi, y1 + y2,
n∑
i=3
yi
}}
.
After orbiting the centralizer invariants to obtain G-invariants (see the end of Section 4.1
in [8] for more detail), these yield the description in Definition 3.8 for the cochain κref =∑
(ij)∈Sn
(
κC(ij) + κL(ij)
)
⊗ (ij) supported off the identity. �
3.3 Pre-Drinfeld orbifold algebra maps
By Lemma 2.5 and Remark 2.6, the polynomial degree one elements of Hochschild 2-cohomology
found in Propositions 3.2 and 3.4 provide a description of all pre-Drinfeld orbifold algebra maps.
Corollary 3.5. The pre-Drinfeld orbifold algebra maps for Sn (n ≥ 3) acting by the doubled
permutation representation on V = W ∗ ⊕W ∼= C2n are the linear 2-cochains κL = κL1 + κLref
for κL1 described in terms of the parameters a1, . . . , a7, b1, . . . , b7 as in Definition 3.7 and κLref
controlled by the parameters a, a⊥, b, b⊥ as in Definition 3.8.
In Theorems 4.1 and 5.9 we will characterize when the maps κL1 and κLref lift separately
to Drinfeld orbifold algebra maps and in Theorem 5.10 we will show it is also possible to lift
κL1 +κLref . Any two lifts of a particular pre-Drinfeld orbifold algebra map must differ by a constant
2-cochain that satisfies the mixed Jacobi identity. Lemma 2.5 and the results in this section
yield the following corollary describing these maps.
Corollary 3.6. For Sn (n ≥ 3) acting on V = W ∗ ⊕W ∼= C2n by the doubled permutation
representation, the Sn-invariant constant 2-cochains satisfying the mixed Jacobi identity are the
maps κC = κC1 + κCref with κC1 given in terms of parameters α and β in Definition 3.7 and κCref
described using parameter c in Definition 3.8.
Degree-One Rational Cherednik Algebras for the Symmetric Group 11
3.4 Definitions of linear and constant cochains
For convenience we collect here the definitions of the components of the maps determined in Pro-
positions 3.2 and 3.4 and that will be needed to lift κL1 in Section 4 and κLref in Section 5.
Some parts of the descriptions below involve sums of basis vectors over subsets of [n] =
{1, . . . , n}. For I ⊆ [n] let vI =
∑
i∈I vi, where v stands for x or y and at times we omit the
set braces in I. Let v⊥I denote the complementary vector v[n] − vI . In all three definitions, Sn
(n ≥ 3) acts by the doubled permutation representation on V = W ∗ ⊕W ∼= C2n equipped with
basis Bx ∪ By = {x1, . . . , xn, y1, . . . , yn}.
Definition 3.7 (cochains supported only on the identity). Given complex parameters ak, bk for
1 ≤ k ≤ 7 and α, β in C, let κL1 :
∧2 V → V and κC1 :
∧2 V → C be the Sn-invariant maps
defined by
κL1 (xi, xj) = a1(xi − xj) + b1(yi − yj), (3.3)
κL1 (yi, yj) = a2(xi − xj) + b2(yi − yj), (3.4)
κL1 (xi, yi) = a3xi + a4x[n] + b3yi + b4y[n], (3.5)
κL1 (xi, yj) = a5xi + a6xj + a7x[n] + b5yi + b6yj + b7y[n], (3.6)
and
κC1 (xi, xj) = κC1 (yi, yj) = 0, κC1 (xi, yi) = α, κC1 (xi, yj) = β,
where 1 ≤ i 6= j ≤ n.
Definition 3.8 (cochains supported only on transpositions). Let a, a⊥, b, b⊥, c be complex
parameters and let T be the set of transpositions in Sn. Define a linear 2-cocycle κLref =∑
g∈T κ
L
g g, where for g = (rs), the component κLg :
∧2 V → V is defined for 1 ≤ i, j ≤ n by
κLg (xi, xj) = κLg (yi, yj) = 0
and
κLg (xi, yj) =
axr,s + a⊥x⊥r,s + byr,s + b⊥yr,s if i = j is in {r, s},
−(axr,s + a⊥x⊥r,s + byr,s + b⊥yr,s) if {i, j} = {r, s},
0 otherwise.
Similarly, the g = (rs) component of the constant 2-cocycle κCref =
∑
g∈T κ
C
g g is defined for
1 ≤ i, j ≤ n by
κCg (xi, xj) = κCg (yi, yj) = 0
and
κCg (xi, yj) =
c if i = j is in {r, s},
−c if {i, j} = {r, s},
0 otherwise.
Lastly, we define a constant 2-cochain κC3-cyc which we use to lift κLref in Section 5.1. The map
κC3-cyc is not a Hochschild 2-cocycle but rather is based on the form of φ(κLref , κ
L
ref) in Proposi-
tions 5.3 and 5.4 and is constructed to ensure φ(κLref , κ
L
ref) = 2ψ(κC3-cyc) as in Proposition 5.5,
thereby clearing the first obstruction to lifting κLref . The cochain κC3-cyc will also clear the second
obstruction to lifting κLref , as verified in Lemma 5.6.
12 B. Foster-Greenwood and C. Kriloff
Definition 3.9 (cochains supported only on 3-cycles). Define an Sn-invariant map κC3-cyc =∑
g∈Sn
κCg g with component maps κCg :
∧2 V → C. If g is not a 3-cycle, let κCg ≡ 0. For a 3-
cycle g = (i j k), define the outcome of κCg on a pair of basis vectors to be zero unless the indices
are two distinct elements of {i, j, k}, in which case the outcome is defined by the following (and
skew-symmetry):
κCg (xi, xj) = κCg (xj , xk) = κCg (xk, xi) =
(
b⊥ − b
)2
and
κCg (yi, yj) = κCg (yj , yk) = κCg (yk, yi) =
(
a⊥ − a
)2
and
κCg (xi, yj) = κCg (yj , xk) = κg(xk, yi) = κCg (yi, xj) = κCg (xj , yk) = κg(yk, xi)
= −
(
a⊥ − a
)(
b⊥ − b
)
.
4 Lie orbifold algebra maps that deform S(W ∗ ⊕ W )#Sn
In Section 3, as summarized in Proposition 3.2 and Definition 3.7, we determined the pre-Drinfeld
orbifold algebra maps κL1 supported only on the identity. Here we find conditions under which
these maps also endow V with a Lie algebra structure — i.e., under which they lift to Lie orbifold
algebra maps because there exists a constant 2-cochain κC such that κ = κL1 + κC also satisfies
the remaining properties (2.4) and (2.5).
Our main goal is to write down conditions on the parameters involved in the definitions
of κL1 , κC1 , and κCref such that properties (2.4) and (2.5) hold, or in other words, such that
φ
(
κL1 , κ
L
1
)
= 2ψ
(
κC1 + κCref
)
and φ
(
κC1 + κCref , κ
L
1
)
= 0. Since 2ψ
(
κC1 + κCref
)
= 0, we have that
κC1 +κCref clears both the first and second obstructions and the map κL1 gives rise to a Lie orbifold
algebra if and only if φ
(
κL1 , κ
L
1
)
= φ
(
κC1 + κCref , κ
L
1
)
= 0. We use this to arrive at characterizing
PBW conditions on parameters as summarized in the proof of Theorem 4.1, which states that κL1
can be lifted to κ = κL1 +κC1 +κCref precisely when a list of 22 homogeneous quadratic conditions
in 17 parameters hold.
It will be convenient along the way to also consider φ
(
κLref , κ
L
1
)
, for use in Theorem 5.10,
by using ∗ to denote either C or L and x to denote either a transposition or the identity and
computing, for v1, v2, v3 ∈ V ,
φ∗x,1(v1, v2, v3) := κ∗x
(
v1, κ
L
1 (v2, v3)
)
+ κ∗x
(
v2, κ
L
1 (v3, v1)
)
+ κ∗x
(
v3, κ
L
1 (v1, v2)
)
as uniformly as possible. This notation omits a factor of two (and hence differs from that in [8])
because ψ(κC1 +κCref) = 0 means the factor of 2 is irrelevant to clearing the first obstruction and
it is also irrelevant to clearing the second obstruction.
First note that due to bilinearity and skew-symmetry it suffices to compute φ∗x,1, with x equal
to the identity or a transposition, on basis triples of six main types for 1 ≤ i, j, k ≤ n, where
n ≥ 3.
1. All basis vectors in W or in W ∗ and i, j, k distinct: (xi, xj , xk), (yi, yj , yk).
2. Two basis vectors in W or in W ∗ and i, j, k distinct: (xi, xj , yk), (yi, yj , xk).
3. Two basis vectors in W or W ∗ and i, j distinct: (xi, xj , yj), (yi, yj , xj).
This is done in the next three subsections.
Degree-One Rational Cherednik Algebras for the Symmetric Group 13
4.1 All basis vectors in W or in W ∗ and three distinct indices
For any distinct indices i, j, k with 1 ≤ i, j, k ≤ n, we have
φ∗x,1(xi, xj , xk) = κ∗x
(
xi, κ
L
1 (xj , xk)
)
+ κ∗x
(
xj , κ
L
1 (xk, xi)
)
+ κ∗x
(
xk, κ
L
1 (xi, xj)
)
.
Using bilinearity, skew-symmetry, and Definitions 3.7 and 3.8 of κ1 and κref yields for x either
the identity or any transposition,
κ∗x(xi, xj − xk) + κ∗x(xj , xk − xi) + κ∗x(xk, xi − xj)
= 2[κ∗x(xi, xj) + κ∗x(xj , xk) + κ∗x(xk, xi)] = 0,
and
κ∗x(xi, yj − yk) + κ∗x(xj , yk − yi) + κ∗x(xk, yi − yj) = 0.
Combining these shows that φ∗x,1(xi, xj , xk) = 0 and similarly φ∗x,1(yi, yj , yk) = 0, for any (dis-
tinct i, j, k with) 1 ≤ i, j, k ≤ n, for x either the identity or a transposition, and with ∗ = C
or ∗ = L. Thus this case imposes no restrictions on any parameters.
4.2 Two basis vectors in W or W ∗ and three distinct indices
For any distinct indices i, j, k with 1 ≤ i, j, k ≤ n, using the definition of κL1 , bilinearity, and
skew-symmetry yields
φ∗x,1(xi, xj , yk) = 2a5κ
∗
x(xi, xj) + a6κ
∗
x(xi − xj , xk) + a7κ
∗
x
(
xi − xj , x[n]
)
− b1κ∗x(yi − yj , yk) + b5 (κ∗x(xi, yj)− κ∗x(xj , yi))
+ (b6 − a1)κ∗x(xi − xj , yk) + b7κ
∗
x
(
xi − xj , y[n]
)
.
When x = 1 and ∗ = C, since κC1 (v, w) = 0 when v, w ∈W or v, w ∈W ∗, we have
φC1,1(xi, xj , yk) = b5(β − β) + (b6 − a1)(β − β) + b7 (α− α+ (n− 1)(β − β)) = 0,
and when x = 1 and ∗ = L using the definition of κL1 yields
φL1,1(xi, xj , yk) = γ1(xi − xj) + γ2(yi − yj),
where
γ1 = a1(a5 + a6 + na7)− b1a2 − b5a6 + a5(b5 + b6 + nb7) + b7(a3 − a5 − a6),
γ2 = b1(a5 + a6 + na7)− b1b2 + b1a5 + b5(b5 − a1 + nb7) + b7(b3 − b5 − b6).
When x = g is a transposition, by Definition 3.8 we have that
φ∗g,1(xi, xj , yk) = (b6 − a1)κ∗g(xi − xj , yk)
=
{
±(b6 − a1)κ∗g(xl, yk), if g = (lk) with l = i or l = j respectively,
0, otherwise,
and we define
γ3 = −(b6 − a1)κC(jk)(xj , yk) = c(b6 − a1).
14 B. Foster-Greenwood and C. Kriloff
Interchanging the roles of x and y and recomputing yields that for distinct i, j, k with
1 ≤ i, j, k ≤ n,
φ∗x,1(yi, yj , xk) = −2b6κ
∗
x(yi, yj)− b5κ∗x(yi − yj , yk)− b7κ∗x
(
yi − yj , y[n]
)
− a2κ∗x(xi − xj , xk)− a6
(
κ∗x(xi, yj)− κ∗x(xj , yi)
)
+ (b2 + a5)κ
∗
x(xk, yi − yj) + a7κ
∗
x
(
x[n], yi − yj
)
.
In particular,
φC1,1(yi, yj , xk) = 0 and φL1,1(yi, yj , xk) = γ4(xi − xj) + γ5(yi − yj),
where
γ4 = −a2(b6 + b5 + nb7)− a2a1 − a2b6 + a6(a6 + b2 + na7) + a7(a3 − a6 − a5),
γ5 = −b2(b6 + b5 + nb7)− a2b1 − a6b5 + b6(a6 + a5 + na7) + a7(b3 − b6 − b5).
When x = g is a transposition, we have
φ∗g,1(yi, yj , xk) = (b2 + a5)κ
∗
g(xk, yi − yj)
=
{
±(b2 + a5)κ
∗
g(xk, yl) if g = (lk) with l = i or l = j respectively,
0 otherwise,
and we define
γ6 = −(b2 + a5)κ
C
g (xk, yj) = c(b2 + a5).
4.3 Two basis vectors in W or W ∗ and two distinct indices
For any distinct indices i, j with 1 ≤ i, j ≤ n, we have
φ∗x,1(xi, xj , yj) = (a3 + a5)κ
∗
x(xi, xj)− b1κ∗x(yi, yj) + a4κ
∗
x(xi, x[n])− a7κ∗x(xj , x[n])
+ (−a1 + b3)κ
∗
x(xi, yj)− b5κ∗x(xj , yi) + (a1 − b6)κ∗x(xj , yj)
+ b4κ
∗
x(xi, y[n])− b7κ∗x(xj , y[n]).
In particular if x = 1 and ∗ = C then we set
γ7 = φC1,1(xi, xj , yj) = α(a1 − b6 + b4 − b7)− β(a1 − b3 + b5 − (n− 1)(b4 − b7)),
and if x = 1 and ∗ = L then
φL1,1(xi, xj , yj) = γ8x[n] + γ9xi + γ10xj + γ11y[n] + γ12yi − γ13yj ,
where
γ8 = a7(b3 − b5)− a4b6 + (b4 − b7)(a4 + (n− 1)a7 + a6),
γ9 = a1(a3 + na4) + a5(b3 + nb4) + b4(a3 − a5 − a6)− a2b1 − b5a6,
γ10 = −a1(a5 + a6 + na7)− a5(b5 + nb7)− b7(a3 − a5 − a6) + a2b1 + b3a6 − a3b6,
γ11 = b7(b3 − b5)− b4b6 + (b4 − b7)(a1 + b4 + (n− 1)b7 + b6)− b1(a4 − a7),
γ12 = b1(a5 − b2 + a3 + na4)− b5(a1 + b6 − b3 − nb4) + b4(b3 − b5 − b6),
γ13 = b1(−b2 + a5 + a3 + na7) + b5(b5 + nb7) + b7(b3 − b5 − b6)− a1(b3 − b6).
Degree-One Rational Cherednik Algebras for the Symmetric Group 15
For x = g a transposition, we have that
φ∗g,1(xi, xj , yj) = (−a1 + b3)κ
∗
g(xi, yj)− b5κ∗g(xj , yi) + (a1 − b6)κ∗g(xj , yj)
=
{
(2a1 − b3 + b5 − b6)κ∗g(xj , yj), if g = (ij),
0, otherwise,
and we define
γ14 = (2a1 − b3 + b5 − b6)κCg (xj , yj) = c(2a1 − b3 + b5 − b6).
Interchanging the roles of x and y and recomputing yields that for any distinct indices i and j
with 1 ≤ i, j ≤ n,
φ∗x,1(yi, yj , xj) = −(b3 + b6)κ
∗
x(yi, yj)− a2κ∗x(xi, xj)− b4κ∗x(yi, y[n]) + b7κ
∗
x(yj , y[n])
+ (a3 + b2)κ
∗
x(xj , yi)− a6κ∗x(xi, yj)− (a5 + b2)κ
∗
x(xj , yj)
+ a4κ
∗
x(x[n], yi)− a7κ∗x(x[n], yj).
In particular, we set
γ15 = φC1,1(yi, yj , xj) = α(−b2 − a5 + a4 − a7) + β(b2 + a3 − a6 + (n− 1)(a4 − a7)),
and
φL1,1(yi, yj , xj) = γ16x[n] + γ17xi + γ18xj + γ19y[n] + γ20yi + γ21yj ,
where
γ16 = a7(a3 − a6)− a4a5 + (a4 − a7)(−b2 + a4 + (n− 1)a7 + a5) + a2(b4 − b7),
γ17 = −a2(a1 + b6 + b3 + nb4) + a6(b2 − a5 + a3 + na4) + a4(a3 − a5 − a6),
γ18 = a2(a1 + b6 + b3 + nb7)− a6(a6 + na7)− a7(a3 − a5 − a6)− b2(a3 − a5),
γ19 = b7(a3 − a6)− b4a5 + (a4 − a7)(b4 + (n− 1)b7 + b5),
γ20 = −b2(b3 + nb4) + b6(a3 + na4) + a4(b3 − b5 − b6)− a2b1 − a6b5,
γ21 = b2(b5 + b6 + nb7)− b6(a6 + na7)− a7(b3 − b5 − b6) + a2b1 + a3b5 − b3a5.
Lastly, when x = g is a transposition, we have
φ∗g,1(xi, xj , yj) = (a3 + b2)κ
∗
g(xj , yi)− a6κ∗g(xi, yj)− (a5 + b2)κ
∗
g(xj , yj)
=
{
−(2b2 + a3 + a5 − a6)κ∗g(xj , yj), if g = (ij),
0, otherwise,
and we define
γ22 = (2b2 + a3 + a5 − a6)κ∗g(xj , yj) = c(2b2 + a3 + a5 − a6).
4.4 Lie orbifold algebra maps
We now use the calculations in Sections 4.1–4.3 to describe all Drinfeld orbifold algebra maps
with linear part supported only on the identity, i.e., all Lie orbifold algebra maps κL1 + κC .
The corresponding Lie orbifold algebras are described in Theorem 7.1.
16 B. Foster-Greenwood and C. Kriloff
Theorem 4.1. Let Sn (n ≥ 3) act on V = W ∗ ⊕ W ∼= C2n by the doubled permutation
representation, and let κL1 and κC1 be as described in Definition 3.7 and κCref be as in Definition 3.8
with complex parameters a1, . . . , a7, b1, . . . , b7, α, β, and c. The Lie orbifold algebra maps are
precisely the maps of the form κ = κL1 +κC1 +κCref satisfying the conditions γi = 0 for 1 ≤ i ≤ 22.
Proof. Let κL be a pre-Drinfeld orbifold algebra map supported only on the identity. By Corol-
lary 3.5 we know κL = κL1 is as given in terms of ai and bi for 1 ≤ i ≤ 7 in Definition 3.7.
In considering property (2.4) when g = 1, note that ψ1 ≡ 0 for any κC , so we must have
φ1,1 ≡ 0 as well. The result of computing φ
(
κL1 , κ
L
1
)
is given in Sections 4.1–4.3 as the values
of the various φL1,1(u, v, w). These show that φ1,1 ≡ 0 precisely when the parameters ai, bi for
i = 1, . . . , 7 satisfy the conditions
γ1 = γ2 = γ4 = γ5 = 0 and γi = 0 for 8 ≤ i ≤ 13, and 16 ≤ i ≤ 21. (4.1)
Since κL is supported only on the identity we must also consider property (2.4) for g 6= 1 and
find all G-invariant constant 2-cochains such that ψ
(
κC
)
= 0 (i.e., satisfying the mixed Jacobi
identity). By Corollary 3.6, these are supported on the identity and transpositions and are given
by κC = κC1 + κCref with κC1 defined using α and β as in Definition 3.7 and κCref defined using c
as in Definition 3.8.
Now assume κC1 + κCref clears the first obstruction, i.e., that the conditions in (4.1) do hold
for κL1 , and consider property (2.5) for each of κC1 and κCref . Using the values of φC1,1(u, v, w)
and φCg,1(u, v, w) in Sections 4.1–4.3 we see that κC1 clears the second obstruction for κL1 , i.e.,
φ
(
κC1 , κ
L
1
)
= 0, precisely when in addition
γ7 = γ15 = 0. (4.2)
We also see that κCref clears the second obstruction for κL1 , i.e., that φ
(
κCref , κ
L
1
)
= 0, precisely
when in addition,
γ3 = γ6 = γ14 = γ22 = 0. (4.3)
Thus κL1 lifts to κL1 + κC1 + κCref if and only if γi = 0 for 1 ≤ i ≤ 22. �
Corollary 4.2. Theorem 4.1 includes precisely the following special cases:
(1) κ = κL1 satisfying conditions (4.1),
(2) κ = κL1 + κC1 satisfying conditions (4.1) and (4.2), and
(3) κ = κL1 + κC1 + κCref with κCref 6≡ 0 satisfying conditions (4.4)–(4.17).
Proof.
Cases (1) and (2). These are immediate by the forms of (4.2) and (4.3) since κC1 ≡ 0 if and
only if α = β = 0 and κCref ≡ 0 if and only if c = 0.
Case (3). Suppose instead that κL1 lifts to κL1 + κC1 + κCref with κCref 6≡ 0. Then in addition
to (4.1) and (4.2), we have c 6= 0. This reduces the conditions in (4.3) to
a1 = b6 and a1 − b3 + b5 = 0, or equivalently b3 − b5 − b6 = 0, (4.4)
b2 = −a5 and b2 + a3 − a6 = 0, or equivalently a3 − a5 − a6 = 0. (4.5)
Degree-One Rational Cherednik Algebras for the Symmetric Group 17
These in turn allow simplification of the conditions for κC1 + κCref to clear the first obstruction
if κCref 6≡ 0 clears the second obstruction for κL1 by reducing (4.1) to
a1(a4 − a7)− (b4 − b7)(a4 + a6 + (n− 1)a7) = 0, (4.6)
b1(a4 − a7)− (b4 − b7)(b4 + b6 + (n− 1)b7) = 0, (4.7)
a2(b4 − b7)− (a4 − a7)(a4 + a5 + (n− 1)a7) = 0, (4.8)
b2(b4 − b7)− (a4 − a7)(b4 + b5 + (n− 1)b7) = 0, (4.9)
a1(a3 + na4) + a5(b3 + nb4)− b1a2 − b5a6 = 0, (4.10)
a1(a3 + na7) + a5(b3 + nb7)− b1a2 − b5a6 = 0, (4.11)
b1(a3 + na4) + b5(b3 + nb4)− 2b1b2 − 2b5b6 = 0, (4.12)
b1(a3 + na7) + b5(b3 + nb7)− 2b1b2 − 2b5b6 = 0, (4.13)
−a2(b3 + nb4) + a6(a3 + na4)− 2a1a2 − 2a5a6 = 0, (4.14)
−a2(b3 + nb7) + a6(a3 + na7)− 2a1a2 − 2a5a6 = 0. (4.15)
Conditions (4.4) and (4.5) also reduce the conditions in (4.2) for κC1 to clear the second obstruc-
tion for κL1 assuming κCref 6≡ 0 also clears the second obstruction for κL1 to
(α+ (n− 1)β)(b4 − b7) = 0, (4.16)
(α+ (n− 1)β)(a4 − a7) = 0. (4.17)
Thus κL1 lifts to κL1 + κC1 + κCref with κCref 6≡ 0 when (4.4)–(4.17) hold. �
Remark 4.3. Note in Case (3) of Corollary 4.2 with c 6= 0 that if α 6= −(n − 1)β, then
a4 = a7 and b4 = b7. These combined with a3 = a5 + a6 and b3 = b5 + b6 in (4.4)–(4.5)
mean that the definitions of κL1 (xi, yj) in (3.6) (but with j allowed to be i) and the definition
of κL1 (xi, yi) in (3.5) agree. Furthermore, conditions (4.6)–(4.9) then hold and conditions (4.10)–
(4.15), simplify further to
a1(a3 + na4) + a5(b3 + nb4)− b1a2 − b5a6 = 0,
b1(a3 + na4) + b5(b3 + nb4)− 2b1b2 − 2b5b6 = 0,
−a2(b3 + nb4) + a6(a3 + na4)− 2a1a2 − 2a5a6 = 0.
4.5 Algebraic varieties corresponding to image constraints
The homogeneous quadratic PBW conditions γi = 0 for 1 ≤ i ≤ 22 give rise to a projective vari-
ety that controls the parameter space for the family of maps in Theorem 4.1 and corresponding
Lie orbifold algebras in Theorem 7.1. Based on computations done for a few specific values of n
in Macaulay2 [14] with the graded reverse lexicographic monomial ordering and the parameter
order a1, . . . , a7, b1, . . . , b7, α, β, c we conjecture that this projective variety has dimension seven.
It contains the lattice of subvarieties described in Table 1 arising from the PBW conditions and
additional constraints on how imκL1 relates to subrepresentations of the doubled permutation
representation. While not needed for Theorem 4.1, these are included as being of potential
independent interest.
Remark 4.4. Observe that xi = x̄i + 1
nx[n] and yi = ȳi + 1
ny[n] can be used in (3.3)–(3.6) to
18 B. Foster-Greenwood and C. Kriloff
decompose the results according to V ∼= h∗ ⊕ ι∗ ⊕ h⊕ ι:
κL1 (xi, xj) = a1(x̄i − x̄j) + b1(ȳi − ȳj), (4.18)
κL1 (yi, yj) = a2(x̄i − x̄j) + b2(ȳi − ȳj), (4.19)
κL1 (xi, yi) = a3x̄i +
1
n
(a3 + na4)x[n] + b3ȳi +
1
n
(b3 + nb4)y[n], (4.20)
κL1 (xi, yj) = a5x̄i+ a6x̄j+
1
n
(a5+ a6+ na7)x[n]+ b5ȳi+ b6ȳj+
1
n
(b5+ b6+ nb7)y[n]. (4.21)
Using Remark 4.4 we see that if the image of κL1 is contained within h∗⊕h then the coefficients
of x[n] and y[n] are zero; i.e.,
a3 + na4 = 0, b3 + nb4 = 0, (4.22)
a5 + a6 + na7 = 0, b5 + b6 + nb7 = 0. (4.23)
If the image of κL1 contains at most one copy of the trivial representation, i.e., some linear
combination of x[n] and y[n], then the coefficients of x[n] and y[n] satisfy this weaker condition:
(a3 + na4)(b5 + b6 + nb7) = (b3 + nb4)(a5 + a6 + na7). (4.24)
Similarly, if the image of κL1 contains at most one copy of the standard representation then the
coefficients of x̄i and ȳi satisfy these conditions:
aibj = biaj for 1 ≤ i < j ≤ 6 and i, j 6= 4. (4.25)
Lastly, if the image of κL1 is contained within ι∗ ⊕ ι then the coefficients of x̄i and ȳi are zero:
ai = bi = 0 for 1 ≤ i ≤ 6 and i 6= 4. (4.26)
In Table 1 we list constraints on imκL1 , resulting conditions, in addition to (4.1)–(4.3), needed
to describe the subvariety of maps subject to each constraint, and the conjectured dimension and
degree based on computations. Dropping condition (4.24) in the first three rows, i.e., allowing
an additional summand of the trivial representation in imκL1 in those cases, did not change the
computed dimension or degree of the variety.
Table 1. Conjectured dimension and degree of projective varieties for Lie orbifold algebras obtained
from constraints on imκL1 . When imκL1 is contained in the trivial representation, the resulting seven
defining polynomials, the dimension and degree, and related simple κ maps can be found by hand.
Constraint on imκL1 Conditions Dimension Degree
contained in perm⊕ std (4.24) 7 8
contained in perm (4.24), (4.25) 6 30
contained in triv (4.24), (4.25), (4.26) 4 1
contained in std⊕ std (4.22), (4.23), (4.24) 6 6
contained in std (4.22), (4.23), (4.24), (4.25) 5 4
5 Drinfeld orbifold algebra maps that deform S(W ∗ ⊕ W )#Sn
In Section 3 we defined a pre-Drinfeld orbifold algebra map κLref . Here we aim to lift κLref to
a Drinfeld orbifold algebra map. We evaluate φ
(
κLref , κ
L
ref
)
and clear the first obstruction by
Degree-One Rational Cherednik Algebras for the Symmetric Group 19
defining a G-invariant 2-cochain κC3-cyc such that φ
(
κLref , κ
L
ref
)
= 2ψ
(
κC3-cyc
)
. We then clear the
second obstruction by showing φ
(
κCref+κ
C
3-cyc, κ
L
ref
)
= 0 and giving conditions on the parameters α
and β for κC1 and a, a⊥, b, b⊥ for κLref such that φ
(
κC1 , κ
L
ref
)
= 0. It follows in Theorem 5.9 that
κLref + κCref + κC3-cyc is always a Drinfeld orbifold algebra map and κLref + κC1 + κCref + κC3-cyc is
a Drinfeld orbifold algebra map precisely when conditions (5.3) and (5.4) hold.
Characterizing in general when κL1 + κLref lifts is straightforward but rather more involved.
Instead, in Theorem 5.10 we provide a nontrivial choice of parameters and verify in that case
that it is possible to lift simultaneously to κL1 + κLref + κC1 + κCref + κC3-cyc. There could very well
be other parameter choices for successfully lifting κL1 + κLref . This is indicated by the question
marks in Table 2 summarizing the results in this and the previous section.
5.1 Clearing obstructions to deformations of S(W ∗ ⊕ W )#Sn
We begin by recalling a lemma from [8] that allows for a reduction in the computations necessary
to remove obstructions and lift κ maps.
Invariance relations. Recall that a cochain µ =
∑
g∈G µgg with components µg :
∧k V → S(V )
is G-invariant if and only if hµg = µhgh−1 for all g, h ∈ G. Equivalently,
h(µg(v1, . . . , vk)) = µhgh−1
(
hv1, . . . ,
hvk
)
for all g, h ∈ G and v1, . . . , vk ∈ V . Thus a G-invariant cochain is determined by its components
for a set of conjugacy class representatives.
In the following lemma that applies to any finite group, one can let µ = κL or µ = κC
and let ν = κL to see that if κL and κC are G-invariant, then φ
(
κ∗, κL
)
and ψ(κ∗) are also G-
invariant. This is helpful because, for instance, if φg = 2ψg for some g ∈ G, then acting by h ∈ G
on both sides shows φhgh−1 = 2ψhgh−1 also. Thus if φg = 2ψg for all g in a set of conjugacy
class representatives, then φ
(
κL, κL
)
= 2ψ
(
κC
)
. Similar reasoning applies to properties (2.3)
and (2.5) of a Drinfeld orbifold algebra map.
Lemma 5.1 ([8, Lemma 5.1]). Let G be a finite group acting linearly on V ∼= Cn. If µ and ν
are G-invariant 2-cochains with ν linear and µ linear or constant, then φ(µ, ν) and ψ(µ) are
G-invariant. Specifically, at the component level, for all x, y, h ∈ G and v1, v2, v3 ∈ V we have
h(φx,y(v1, v2, v3)) = φhxh−1,hyh−1
(
hv1,
hv2,
hv3
)
.
5.2 Clearing the first obstruction
We begin by recording simplifications of a summand, φ∗σ,τ , of the component φ∗g of φ
(
κ∗ref , κ
L
ref
)
,
where ∗ stands for L or C. Simplification of φ∗σ,τ (u, v, w) depends on the location of the basis
vectors relative to W and W ∗ and relative to the fixed spaces V σ and V τ , so recall that v∗ is
the vector dual to v and define the following indicator function. For g ∈ Sn and v ∈ V , let
δg(v) =
{
1 if v ∈ V g,
0 otherwise.
Note that for g ∈ G and v ∈ V ,
δg(v
∗) = δg(v).
Remark. Let φ∗σ,τ be as in Lemma 5.2. Then for all u, v, w ∈ V we have
φ∗σ,τ (τu, τv, τw) = φ∗σ,τ (u, v, w). (5.1)
This follows from the definition of φ∗σ,τ and that κLτ is τ -invariant.
20 B. Foster-Greenwood and C. Kriloff
Lemma 5.2. Let κ∗ref with ∗ = L or ∗ = C be as in Definition 3.8, with common parameters
a, a⊥, b, b⊥ ∈ C. Denote a term of the component φ∗g of φ
(
κ∗ref , κ
L
ref
)
by φ∗σ,τ , where σ and τ are
transpositions such that στ = g.
(1) If u, v, w ∈W , u, v, w ∈W ∗, u, v ∈ V τ , or u ∈ V τ ∩ V σ, then φ∗σ,τ (u, v, w) = 0.
(2) If u ∈ V τ \ V σ and v /∈ V τ , then the basis vectors moved by τ are of the form v, τv,
v∗, τv∗. We have φ∗σ,τ (v, τv, u∗) = 0,
φ∗σ,τ (u, v, v∗) = −φ∗σ,τ (u, v, τv∗) =
{
2
(
a⊥ − a
)
[1− δτ (σu)]κ∗σ(u∗, u) if u ∈W,
2
(
b⊥ − b
)
[1− δτ (σu)]κ∗σ(u, u∗) if u ∈W ∗,
and by (5.1),
φ∗σ,τ (u, τv, τv∗) = −φ∗σ,τ (u, τv, v∗) = φ∗σ,τ (u, v, v∗).
(3) If v /∈ V τ , then
φ∗σ,τ (v, τv, v∗) =
{
2
(
a⊥ − a
)[
δτ (σv)κ∗σ(v∗, v) + δτ (στv)κ∗σ(τv∗, τv)
]
if v ∈W,
2
(
b⊥ − b
)[
δτ (σv)κ∗σ(v, v∗) + δτ (στv)κ∗σ(τv, τv∗)
]
if v ∈W ∗,
and by (5.1),
φ∗σ,τ (v, τv, τv∗) = −φ∗σ,τ (v, τv, v∗).
Proof.
Case (1). When u, v, w ∈W or u, v, w ∈W ∗, then φ∗σ,τ (u, v, w) = 0 because
φ∗σ,τ (u, v, w) = κ∗σ
(
u+ τu, κLτ (v, w)
)
+ κ∗σ
(
v + τv, κLτ (w, u)
)
+ κ∗σ
(
w + τw, κLτ (u, v)
)
, (5.2)
and κLτ is zero whenever both input vectors are in W ∗ or both are in W . As in [8], φ∗σ,τ (u, v, w) =
0 when u, v ∈ V τ follows from (5.2) and that V τ ⊆ kerκ∗τ , while φ∗σ,τ (u, v, w) = 0 when u ∈
V τ ∩ V σ uses also V σ ⊆ kerκ∗σ.
Case (2). Assume u ∈ V τ \ V σ and v /∈ V τ . First note that
φ∗σ,τ (v, τv, u∗) = 0
by using (5.1) and the alternating property to see that φ∗σ,τ (v, τv, u∗) = −φ∗σ,τ (v, τv, u∗).
We can reduce to φ∗σ,τ (u, v, v∗) = 2κ∗σ
(
u, κLτ (v, v∗)
)
by using u ∈ V τ ⊆ kerκ∗τ in (5.2).
Using bilinearity and V σ ⊆ kerκ∗σ, the right hand side is a linear combination of expressions
κ∗σ
(
u, hu
)
and κ∗σ
(
u, hu∗
)
for h ∈ 〈σ〉. The appropriate coefficients in terms of a, a⊥, b, b⊥ can
be described in terms of the indicator function for the fixed space of τ and depend on whether
u ∈W ∗ or u ∈W . Also note that
∑
h∈〈σ〉
hu ∈ V σ ⊆ kerκ∗σ. Thus, for u ∈W ∗ we have
κ∗σ
(
u, κLτ (v, v∗)
)
=
∑
h∈〈σ〉
[
a
(
1− δτ
(
hu
))
+ a⊥δτ
(
hu
)]
κ∗σ
(
u, hu
)
+
[
b
(
1− δτ
(
hu∗
))
+ b⊥δτ
(
hu∗
)]
κ∗σ(u, hu∗)
=
(
a⊥ − a
) ∑
h∈〈σ〉
δτ
(
hu
)
κ∗σ
(
u, hu
)
+
(
b⊥ − b
) ∑
h∈〈σ〉
δτ
(
hu∗
)
κ∗σ
(
u, hu∗
)
=
(
b⊥ − b
)[
δτ (u)κ∗σ(u, u∗) + δτ (σu)κ∗σ(u, σu∗)
]
,
Degree-One Rational Cherednik Algebras for the Symmetric Group 21
where the term with coefficient a⊥ − a is zero because κ∗σ(u, u) = κ∗σ(u, σu) = 0. Since u ∈ V τ
and κ∗σ(u, σu∗) = −κ∗σ(u, u∗), this yields
φ∗σ,τ (u, v, v∗) = 2κ∗σ
(
u, κLτ (v, v∗)
)
= 2
(
b⊥ − b
)[
1− δτ (σu)
]
κ∗σ(u, u∗).
When u ∈W , the calculation of κ∗σ
(
u, κLτ (v, v∗)
)
involves a sign difference, and the coefficients
on the two sums are reversed, so the one with coefficient a⊥−a survives and yields φ∗σ,τ (u, v, v∗) =
2
(
a⊥ − a
)[
1− δτ (σu)
]
κ∗σ(u∗, u).
Similar calculations show
φ∗σ,τ (u, v, τv∗) = −φ∗σ,τ (u, v, v∗).
Case (3). Assume v /∈ V τ . Then κLτ (v, τv) = 0 and hence
φ∗σ,τ (v, τv, v∗) = κ∗σ
(
v + τv, κLτ (τv, v∗)
)
+ κ∗σ
(
τv + v, κLτ (v∗, v)
)
= −2κ∗σ
(
v + τv, κLτ (v, v∗)
)
.
A calculation as in case (2), using κLτ (τv, τv∗) = κLτ (v, v∗) and that δτ (v) = δτ (τv) = 0 yields
φ∗σ,τ (v, τv, v∗) = −2
[
κ∗σ
(
v, κLτ (v, v∗)
)
+ κ∗σ
(
τv, κLτ (τv, τv∗)
)]
=
{
2
(
a⊥ − a
)[
δτ (σv)κ∗σ(v∗, v) + δτ (στv)κ∗σ(τv∗, τv)
]
if v ∈W,
2
(
b⊥ − b
)[
δτ (σv)κ∗σ(v, v∗) + δτ (στv)κ∗σ(τv, τv∗)
]
if v ∈W ∗.
�
As mentioned in the outline of the proof of Theorem 5.9, the next two propositions are used
to evaluate both φ
(
κLref , κ
L
ref
)
and φ
(
κCref , κ
L
ref
)
.
Proposition 5.3. Let κ∗ref with ∗ = L or ∗ = C be as in Definition 3.8. For g ∈ Sn where
n ≥ 3, let φ∗g be the g-component of φ
(
κ∗ref , κ
L
ref
)
. If g is not a 3-cycle then φ∗g ≡ 0.
Proof. Since κ∗ref is supported only on transpositions and the only cycle types that arise as
a product of two transpositions are the identity, double transpositions, and 3-cycles, it suffices
to consider only the components φ∗1 and φ∗g where g is a double transposition, and in fact only φ∗1
when n = 3. Since hσhτ = h(στ), it suffices to use only representatives of orbits of factor pairs
under the action of Sn by diagonal conjugation.
Case 1 (g = 1). The identity component φ∗1 of φ(κ∗ref , κ
L
ref) is a sum of terms φ∗σ,σ−1 , where
σ ranges over the set of transpositions in Sn. For each transposition σ, since σ−1 = σ we have
imκLσ−1 = imκLσ ⊆ V σ ⊆ kerκ∗σ, and thus κ∗σ(u, κLσ−1(v, w)) = 0 for all u, v, w ∈ V . It follows
that φ∗σ,σ−1 ≡ 0 for each transposition σ ∈ Sn, and hence, φ∗1 ≡ 0.
Case 2 (g = (12)(34)). Note that φ∗g = φ∗(12),(34) + φ∗(34),(12). Since we know that both
imκL(34) ⊆ V
(12) ⊆ kerκ∗(12) and imκL(12) ⊆ V
(34) ⊆ kerκ∗(34), we see that φ∗g(u, v, w) ≡ 0. �
Proposition 5.4. Let κref = κLref + κCref be as in Definition 3.8 with parameters a, b, c ∈ C, and
let φ∗g denote the g-component of φ
(
κ∗ref , κ
L
ref
)
, where ∗ = L or ∗ = C and g is a 3-cycle. Then
φCg ≡ 0. For φLg we have φLg (u, v, w) = 0 if u ∈ V g, and for v /∈ V g, φLg
(
v, gv, g
2
v
)
= 0, and
φLg (v, gv, v∗) = φLg
(
gv, g
2
v, v∗
)
= φLg
(
g2v, v, v∗
)
=
{
2
(
a⊥ − a
)(
b⊥ − b
)
(gv − v) + 2
(
a⊥ − a
)2
(gv∗ − v∗) if v ∈W ,
2
(
a⊥ − a
)(
b⊥ − b
)
(gv − v) + 2
(
b⊥ − b
)2
(gv∗ − v∗) if v ∈W ∗,
with values on triples involving a third basis vector of the form gv∗ or g2v∗ obtained by acting
by g or g2 respectively.
22 B. Foster-Greenwood and C. Kriloff
Proof. By the orbit property in Lemma 5.1, it suffices to evaluate φ∗g for the conjugacy class
representative g = (123). Note that Z(g) = 〈(123)〉 × Sym{4,...,n}, and the factorizations of g as
a product of transpositions are all in the same Z(g)-orbit under diagonal conjugation, so
φ∗g = φ∗(12),(23) + φ∗(23),(31) + φ∗(31),(12).
For each pair of transpositions σ and τ with στ = g = (123), we have V g ⊆ V σ ∩ V τ and
V σ ∩ V τ ⊆ kerφ∗σ,τ by Lemma 5.2(1). So if any vector in a basis triple is in V g, then φ∗g
evaluates to zero. It remains only to consider triples {u, v, w} ⊆ {x1, x2, x3, y1, y2, y3}.
Lemma 5.2(1) yields φ∗g(x1, x2, x3) = φ∗g(y1, y2, y3) = 0. There are, up to permutation,
1
2
(
6
3
)
− 1 = 9 basis triples with two elements in W ∗ and one element in W :
x1, x2, y1, x2, x3, y1, x3, x1, y1,
x2, x3, y2, x3, x1, y2, x1, x2, y2,
x3, x1, y3, x1, x2, y3, x2, x3, y3.
The G-invariance in Lemma 5.1 yields
φ∗g(
gu, gv, gw) = gφ∗g(u, v, w),
and thus since each of the three columns of basis triples is a g-orbit, it suffices to compute φ∗g
on just the basis triples
x1, x2, y1, x2, x3, y1, and x3, x1, y1.
For each of these three basis triples, u, v, w, the result φ∗g(u, v, w) will be the same by Lemma 5.2
(although the reason varies for a given term on different triples), namely
φ∗g(u, v, w) = φ∗(12),(23)(u, v, w) + φ∗(23),(31)(u, v, w) + φ∗(31),(12)(u, v, w)
= 0− 2
(
b⊥ − b
)
κ∗(23)(x3, y3) + 2
(
b⊥ − b
)
κ∗(31)(x1, y1)
=
0 if *=C,
−2
(
b⊥ − b
)[
ax2,3 + a⊥x⊥2,3 + by2,3 + b⊥y⊥2,3
]
+ 2
(
b⊥ − b
)[
ax1,3 + a⊥x⊥1,3 + by1,3 + b⊥y⊥1,3
]
if ∗ = L
=
{
0 if ∗ = C,
2
(
b⊥ − b
)[(
a⊥ − a
)
(x2 − x1) +
(
b⊥ − b
)
(y2 − y1)
]
if ∗ = L.
A similar reduction and computation applies to the nine basis triples with two elements in W
and one element in W ∗, and that combined with the orbiting properties in Lemma 5.1 lead to
the conclusion in the statement. �
Using the form of φ
(
κLref , κ
L
ref
)
in Propositions 5.3 and 5.4 we define the cochain κC3-cyc in Defi-
nition 3.9 to ensure φ
(
κLref , κ
L
ref
)
= 2ψ
(
κC3-cyc
)
as in the next proposition.
Proposition 5.5. Let κLref and κC3-cyc be as in Definitions 3.8 and 3.9, with common parameters
a, a⊥, b, b⊥ ∈ C. Then φ
(
κLref , κ
L
ref
)
= 2ψ
(
κC3-cyc
)
.
Proof. We compare the component φg of φ
(
κLref , κ
L
ref
)
with the component 2ψg of 2ψ
(
κC3-cyc
)
.
If g is not a 3-cycle, then φg ≡ 0 by Proposition 5.3; and κC3-cyc is not supported on g, so ψg ≡ 0
as well. If g is a 3-cycle, then it suffices to compare components φg and 2ψg on basis triples of
the form in the statement of Proposition 5.4.
Degree-One Rational Cherednik Algebras for the Symmetric Group 23
Case 1. If u ∈ V g, then φg(u, v, w) = 0 by Proposition 5.4 and ψg(u, v, w) = 0 because
gu− u = 0 and V g ⊆ kerκCg .
Case 2. If v /∈ V g, then φg
(
v, gv, g
2
v
)
= 0 by Proposition 5.4 and ψg
(
v, gv, g
2
v
)
= 0 by
g-invariance of κCg .
Case 3. For triples of the form (v, gv, v∗),
(
gv, g
2
v, v∗
)
, and
(
g2v, v, v∗
)
with v ∈ V g, use
Proposition 5.4 to find φg(v,
gv, v∗) = φg
(
gv, g
2
v, v∗
)
= φg
(
g2v, v, v∗
)
and Definition 3.9 to confirm
that φg = 2ψg on each such triple, using that
ψg(v,
gv, v∗) = κCg (v, gv)(gv∗ − v∗) + κCg (gv, v∗)(gv − v) + κCg (v∗, v)
(
g2v − gv
)
,
ψg
(
gv, g
2
v, v∗
)
= κCg
(
gv, g
2
v
)
(gv∗ − v∗) + κCg
(
g2v, v∗
)(
g2v − gv
)
+ κCg (v∗, gv)
(
v − g2v
)
,
ψg
(
g2v, v, v∗
)
= κCg
(
g2v, v
)
(gv∗ − v∗) + κCg (v, v∗)
(
v − g2v
)
+ κCg
(
v∗, g
2
v)(gv − v
)
. �
5.3 Clearing the second obstruction
The final step in determining when the cochain κ = κLref +κC3-cyc +κCref +κC1 is a Drinfeld orbifold
algebra map is to understand when φ
(
κC3-cyc+κCref +κ
C
1 , κ
L
ref
)
= 0, which is stated as Corollary 5.8
and follows immediately from Propositions 5.3 and 5.4 and Lemmas 5.6 and 5.7. This clears the
second obstruction and completes the proof of Theorem 5.9.
Lemma 5.6. Let κLref and κC3-cyc be as in Definitions 3.8 and 3.9, with common parameters
a, a⊥, b, b⊥ ∈ C. Denote a term of the component φg of φ
(
κC3-cyc, κ
L
ref
)
by φσ,τ , where σ is
a 3-cycle and τ is a transposition such that στ = g. Then φσ,τ ≡ 0.
Proof. The proof proceeds by considering the same exhaustive cases as in Lemma 5.2, but
using the definition of κC3-cyc to show in fact φσ,τ ≡ 0 in cases (2) and (3). Showing that
φ∗σ,τ (u, v, w) = 0 when u, v, w ∈ W , u, v, w ∈ W ∗, u, v ∈ V τ , or u ∈ V τ ∩ V σ proceeds exactly
as in the proof of Lemma 5.2 since the methods did not depend on anything about κ∗σ other
than V σ ⊆ kerκ∗σ.
As in the proof of case (2) in Lemma 5.2, assume u ∈ V τ \ V σ and v /∈ V τ and note
φσ,τ (v, τv, u∗) = 0,
and
φσ,τ (u, v, v∗) = 2κCσ
(
u, κLτ (v, v∗)
)
.
Using bilinearity and V σ ⊆ kerκ∗σ, the right hand side is a linear combination of expressions
κ∗σ
(
u, hu
)
for h ∈ 〈σ〉. The appropriate coefficients in terms of a, a⊥, b, b⊥ can be described
in terms of the indicator function for the fixed space of τ and depend on whether u ∈ W ∗ or
u ∈W . Also,
∑
h∈〈σ〉
hu ∈ V σ ⊆ kerκ∗σ. Thus for u ∈W ∗,
κCσ
(
u, κLτ (v, v∗)
)
=
(
a⊥ − a
) ∑
h∈〈σ〉
δτ
(
hu
)
κCσ
(
u, hu
)
+
(
b⊥ − b
) ∑
h∈〈σ〉
δτ
(
hu∗
)
κCσ
(
u, hu∗
)
=
(
a⊥ − a
)[
δτ (σu)κCσ (u, σu) + δτ
(
σ2
u
)
κCσ
(
u, σ
2
u
)]
+
(
b⊥ − b
)[
δτ (σu)κCσ (u, σu∗) + δτ
(
σ2
u
)
κCσ
(
u, σ
2
u∗
)]
=
[
δτ (σu)− δτ
(
σ2
u
)][(
a⊥− a
)(
b⊥− b
)2 − (b⊥− b)(a⊥− a)(b⊥− b)] = 0,
and hence φσ,τ (u, v, v∗) = 0. A similar calculation shows that
φσ,τ (u, v, τv∗) = −φσ,τ (u, v, v∗) = 0
24 B. Foster-Greenwood and C. Kriloff
and applying G-invariance leads to the same conclusions when u ∈W . Then (5.1) also implies
φσ,τ (u, τv, τv∗) = −φσ,τ (u, τv, v∗) = φσ,τ (u, v, v∗) = 0.
For case (3) assume v /∈ V τ and note that as in the proof of Lemma 5.2,
φσ,τ (v, τv, v∗) = −2κCσ
(
v + τv, κLτ (v, v∗)
)
.
Since κCσ
(
τv, κLτ (v, v∗)
)
= κCσ
(
τv, κLτ (τv, τv∗)
)
= κCσ
(
v, κLτ (v, v∗)
)
= 0 by the same calculation as
in case (2) except with τv and v in place of u, it follows that
φσ,τ (v, τv, v∗) = −2
[
κCσ
(
v, κLτ (v, v∗)
)
+ κCσ
(
τv, κLτ (v, v∗)
)]
= 0.
Lastly, (5.1) yields
φσ,τ (v, τv, τv∗) = −φσ,τ (v, τv, v∗) = 0. �
The case where ∗ = L is included in the preliminary calculations of the following lemma
because it will be useful as a starting point in the proof of Theorem 5.10. We note that when
n = 2 conditions (5.3) and (5.4) need to be modified to a(α+ β) = b(α+ β) = 0.
Lemma 5.7. Let κC1 and κLref be as in Definitions 3.7 and 3.8, with parameters α, β ∈ C and
a, a⊥, b, b⊥ ∈ C respectively. Denote a term of the component φg of φ
(
κ∗1, κ
L
ref
)
by φ∗1,g, where
∗ = C or ∗ = L and g is a transposition. Then φC1,g = 0 if and only if the following conditions
hold
αa+ β
(
a+ (n− 2)a⊥
)
= 0, αa⊥ + β
(
2a+ (n− 3)a⊥
)
= 0, (5.3)
αb+ β
(
b+ (n− 2)b⊥
)
= 0, αb⊥ + β
(
2b+ (n− 3)b⊥
)
= 0. (5.4)
Proof. As in Section 4, it suffices to compute φ∗1,g on basis triples of the following forms for
1 ≤ i, j, k ≤ n.
1. All basis vectors in W or in W ∗ and i, j, k distinct: (xi, xj , xk), (yi, yj , yk).
2. Two basis vectors in W or in W ∗ and i, j, k distinct: (xi, xj , yk), (yi, yj , xk).
3. Two basis vectors in W or W ∗ and i, j distinct: (xi, xj , yj), (yi, yj , xj).
Case 1. For any distinct indices i, j, k with 1 ≤ i, j, k ≤ n, using (2.6) and Definition 3.8
of κLref it is immediate that φ∗1,g(xi, xj , xk) = 0, and in similar fashion φ∗1,g(yi, yj , yk) = 0, for any
(distinct i, j, k with) 1 ≤ i, j, k ≤ n. Thus this case imposes no conditions on any parameters.
Case 2. For any distinct indices i, j, k with 1 ≤ i, j, k ≤ n, using the definitions of κLref
and κC1 , bilinearity, and skew-symmetry yields
φ∗1,g(xi, xj , yk) =
2κ∗1
(
xj , axik + a⊥x⊥ik + byik + b⊥y⊥ik
)
if g = (ik),
−2κ∗1
(
xi, axjk + a⊥x⊥jk + byjk + b⊥y⊥jk
)
if g = (jk),
0 otherwise.
=
2
[
αb⊥ + β
(
2b+ (n− 3)b⊥
)]
if g = (ik) and ∗ = C,
−2
[
αb⊥ + β
(
2b+ (n− 3)b⊥
)]
if g = (jk) and ∗ = C,
0 otherwise.
Interchanging the roles of x and y and recomputing yields that for any distinct indices i, j,
k with 1 ≤ i, j, k ≤ n,
φ∗1,g(yi, yj , xk) =
2
[
αa⊥ + β
(
2a+ (n− 3)a⊥
)]
if g = (ik) and ∗ = C,
−2
[
αa⊥ + β
(
2a+ (n− 3)a⊥
)]
if g = (jk) and ∗ = C,
0 otherwise.
Degree-One Rational Cherednik Algebras for the Symmetric Group 25
Case 3. For any distinct indices i, j with 1 ≤ i, j ≤ n, using the definitions of κLref and κC1 ,
bilinearity, and skew-symmetry yields
φ∗1,g(xi, xj , yj) =
2κ∗1
(
xi + xj , axij + a⊥x⊥ij + byij + b⊥y⊥ij
)
if g = (ij),
2κ∗1
(
xi, axjk + a⊥x⊥jk + byjk + b⊥y⊥jk
)
if g = (jk),
0 otherwise.
=
4
[
αb+ β
(
b+ (n− 2)b⊥
)]
if g = (ij) and ∗ = C,
2
[
αb⊥ + β
(
2b+ (n− 3)b⊥
)]
if g = (jk) and ∗ = C,
0 otherwise.
Interchanging the roles of x and y and recomputing yields that for any distinct indices i, j
with 1 ≤ i, j ≤ n,
φ∗1,g(yi, yj , xj) =
4
[
αa+ β
(
a+ (n− 2)a⊥
)]
if g = (ij) and ∗ = C,
2
[
αa⊥ + β
(
2a+ (n− 3)a⊥
)]
if g = (jk) and ∗ = C,
0 otherwise.
Setting the results in cases 2 and 3 equal to zero yields conditions (5.3) and (5.4). �
Corollary 5.8. Let κLref and κC3-cyc be as in Definitions 3.8 and 3.9, with common parameters
a, a⊥, b, b⊥ ∈ C. For every g ∈ Sn, the component φg of φ
(
κCref + κC3-cyc, κ
L
ref
)
is identically zero.
The component φg of φ
(
κC1 , κ
L
ref
)
is zero for all g ∈ Sn if and only if conditions (5.3) and (5.4)
given in Lemma 5.7 hold.
Proof. In fact, each component of φ
(
κCref , κ
L
ref
)
is identically zero by Propositions 5.3 and 5.4,
and each component of φ
(
κC3-cyc, κ
L
ref
)
is identically zero by Lemma 5.6. By Lemma 5.7 we have
φ
(
κC1 , κ
L
ref
)
= 0 if and only if conditions (5.3) and (5.4) are satisfied because the component φg
is identically zero for g not a transposition by the definitions of κC1 and κLref . �
Conditions (5.3) and (5.4) given in Lemma 5.7 give rise to a variety that controls the parame-
ter space for the family of maps in Theorem 5.9 and Drinfeld orbifold algebras in Theorem 7.2.
We conjecture that this projective variety has dimension four based on computations done for
a few specific values of n in Macaulay2 [14] with the graded reverse lexicographic monomial
ordering and the parameter order a, a⊥, b, b⊥, α, β, c.
5.4 Drinfeld orbifold algebra maps
Now we use the details of clearing the obstructions from Section 5.1 to describe all Drinfeld orbi-
fold algebra maps with linear part supported only off the identity. The corresponding Drinfeld
orbifold algebras are given in Theorem 7.2.
Theorem 5.9. For Sn (n ≥ 3) acting on V = W ∗ ⊕ W ∼= C2n by the doubled permutation
representation, the Drinfeld orbifold algebra maps supported only off the identity are precisely
the maps of the form κ = κLref + κC3-cyc + κCref + κC1 , with κLref and κCref as in Definition 3.8, κC3-cyc
as in Definition 3.9, κC1 as in Definition 3.7, and with the parameters a, a⊥, b, b⊥, c, α, and β
satisfying these conditions derived in Lemma 5.7:
(5.3) αa+ β
(
a+ (n− 2)a⊥
)
= 0, αa⊥ + β
(
2a+ (n− 3)a⊥
)
= 0,
(5.4) αb+ β
(
b+ (n− 2)b⊥
)
= 0, αb⊥ + β
(
2b+ (n− 3)b⊥
)
= 0.
In particular, κ = κLref + κC3-cyc + κCref is always a Drinfeld orbifold algebra map.
26 B. Foster-Greenwood and C. Kriloff
Proof. Suppose κL is a pre-Drinfeld orbifold algebra map supported only off the identity.
By Corollary 3.5 we must have κL = κLref for some parameters a, a⊥, b, b⊥ ∈ C as in Definition 3.8.
It remains to find all G-invariant maps κC such that properties (2.4) and (2.5) of a Drinfeld
orbifold algebra map also hold.
First we find a particular lift.
� First obstruction. Propositions 5.3 and 5.4 give the value of φ
(
κLref , κ
L
ref
)
. These values sug-
gest how to construct the Sn-invariant map κC3-cyc such that property (2.4) holds, as given
in Definition 3.9. Proposition 5.5 then verifies that φ
(
κLref , κ
L
ref
)
and 2ψ
(
κC3-cyc
)
are indeed
equal.
� Second obstruction. By Lemma 5.6, we have that φ
(
κC3-cyc, κ
L
ref
)
= 0.
Thus κ = κLref + κC3-cyc is a Drinfeld orbifold algebra map.
Next, we modify this particular lift to obtain all possible lifts. Let κC be any G-invariant
constant 2-cochain.
� First obstruction. Since φ
(
κLref , κ
L
ref) = 2ψ(κC3-cyc
)
, it follows that φ
(
κLref , κ
L
ref
)
= 2ψ
(
κC
)
if and only if ψ
(
κC − κC3-cyc
)
= 0. By Corollary 3.6 this occurs if and only if κC − κC3-cyc =
κCref+κ
C
1 , with κCref as in Definition 3.8 for some parameter c ∈ C and κC1 as in Definition 3.7
for some parameters α, β ∈ C.
� Second obstruction. By Corollary 5.8, φ
(
κC1 + κCref + κC3-cyc, κ
L
ref
)
= 0 if and only if (5.3)
and (5.4) are satisfied. This occurs in particular when κC1 ≡ 0 in which case κCref + κC3-cyc
clears the second obstruction and κLref lifts with no restrictions on the parameters a, a⊥,
b, b⊥, or c.
Thus the lifts of κLref to a Drinfeld orbifold algebra map are precisely the maps of the form
κ = κLref + κC1 + κCref + κC3-cyc satisfying conditions (5.3) and (5.4). �
Lastly, by specifying parameters we obtain some Drinfeld orbifold algebra maps that are
supported both on and off the identity. The proof uses results related to clearing obstructions
that appeared in Section 4 and Section 5.1.
Theorem 5.10. For Sn (n ≥ 3) acting on V = W ∗ ⊕W ∼= C2n by the doubled permutation
representation, there are Drinfeld orbifold algebra maps of the form κ = κL + κC with linear
part κL = κL1 + κLref and constant part κC = κC1 + κCref + κC3-cyc, where
(1) κL1 is as described in Definition 3.7 with ai = bi = 0 for i = 1, 2, 3, 5, 6 and a4 = a7
and b4 = b7 are not both zero,
(2) κC1 is as described in Definition 3.7 with α = β,
(3) κLref and κCref are as in Definition 3.8 and κC3-cyc is as in Definition 3.9 with 2a+(n−2)a⊥ =
2b+ (n− 2)b⊥ = 0, but a, a⊥, b, and b⊥ not all zero.
Proof. As in the proof of Theorem 5.9, even without the given parameter choices we have
φ
(
κLref , κ
L
ref
)
= 2ψ
(
κC3-cyc
)
= 2ψ
(
κC1 + κCref + κC3-cyc
)
. Setting ai = bi = 0 for i = 1, 2, 3, 5, 6,
a4 = a7, and b4 = b7 in (4.1) and in the values of φLg,1(u, v, w) in Sections 4.1–4.3 shows
φ
(
κL1 , κ
L
1
)
= 0 and φ
(
κLref , κ
L
1
)
= 0 respectively. Using the forms of φL1,g given in Lemma 5.7 and
Degree-One Rational Cherednik Algebras for the Symmetric Group 27
the assumptions that a4 = a7 and b4 = b7 yield that
φL1,g(xi, xj , yk) =
2
(
2b+ (n− 2)b⊥
)(
a4x[n] + b4y[n]
)
if g = (ik),
−2
(
2b+ (n− 2)b⊥
)(
a4x[n] + b4y[n]
)
if g = (jk),
0 otherwise,
φL1,g(xi, xj , yj) =
4
(
2b+ (n− 2)b⊥
)(
a4x[n] + b4y[n]
)
if g = (ij),
2
(
2b+ (n− 2)b⊥
)(
a4x[n] + b4y[n]
)
if g = (jk),
0 otherwise,
and similarly for φL1,g(yi, yj , xk) and φL1,g(yi, yj , xj), but replacing b with a and b⊥ with a⊥. By the
hypothesis on a, a⊥, b, and b⊥, all of these are zero, so φ
(
κL1 , κ
L
ref
)
= 0 and hence
φ
(
κL1 + κLref , κ
L
1 + κLref
)
= φ
(
κLref , κ
L
ref
)
= 2ψ
(
κC3-cyc
)
= 2ψ
(
κC1 + κCref + κC3-cyc
)
.
Thus with the given parameter choices κC1 +κCref +κC3-cyc clears the first obstruction for κL1 +κLref .
We claim κC1 + κCref + κC3-cyc also clears the second obstruction for κL1 + κLref because
φ
(
κC1 + κCref + κC3-cyc, κ
L
1 + κLref
)
= 0.
By the assumptions on ai and bi for 1 ≤ i ≤ 7 conditions (4.2) and (4.3) are satisfied and thus
φ
(
κC1 + κCref , κ
L
1
)
= 0. Also φ
(
κC3-cyc, κ
L
1
)
= 0. This is because for any 1 ≤ i, j, k ≤ n and any
3-cycle g, by a1 = a2 = b1 = b2 = 0 we have
φCg,1(xi, xj , xk) = φCg,1(yi, yj , yk) = 0,
and by a4 = a7 and b4 = b7 we have
φCg,1(xi, xj , yk) = κCg
(
xi, κ
L
1 (xj , yk)
)
+ κCg
(
xj , κ
L
1 (yk, xi)
)
= κCg (xi − xj , a4x[n] + b4y[n]) = 0
because κCg
(
xi,
gxi + g−1
xi
)
= 0 regardless of whether i ∈ V g by Definition 3.9, and similarly
φCg,1(yi, yj , xk) = 0. By Corollary 5.8 we know that φ
(
κCref +κC3-cyc, κ
L
ref
)
= 0 in general, and that
φ
(
κC1 , κ
L
ref
)
= 0 because conditions (5.3) and (5.4) are satisfied when α = β and 2a+(n−2)a⊥ =
2b+ (n− 2)b⊥ = 0.
Thus with the given choices of parameters, κC1 + κCref + κC3-cyc clears the second obstruction
as well and lifts κL1 + κLref to a Drinfeld orbifold algebra map. �
6 Drinfeld orbifold algebra maps that deform S(h∗ ⊕ h)#Sn
We now use the results in Sections 4 and 5 on Lie and Drinfeld orbifold algebra maps that
produce deformations of the skew group algebra S(W ∗ ⊕W )#Sn in order to understand which
maps produce deformations of S(h∗ ⊕ h)#Sn.
In contrast to the complicated families of Lie orbifold algebras and maps in Theorems 4.1
and 7.1, when Sn instead acts on its doubled standard subrepresentation h∗⊕h there are no Lie
orbifold algebra maps with nonzero linear part (Theorem 6.3). However, Theorem 6.4 describes
a three-parameter family of Drinfeld orbifold algebra maps that do provide polynomial degree one
deformations generalizing the sln-type rational Cherednik algebras H0,c (see also Theorem 7.3).
We begin with a result that applies to any finite group and provides conditions under which
we can combine Drinfeld orbifold algebra maps for subrepresentations into a map for their
direct sum.
28 B. Foster-Greenwood and C. Kriloff
Proposition 6.1. Let G be a finite group acting linearly on finite-dimensional vector spaces
U1, U2, . . . , Ur. Given Drinfeld orbifold algebra maps κ|Ui for G acting on Ui (i = 1, . . . , r),
define κ on
∧2 (⊕r
i=1 Ui
)
so that κ agrees with κ|Ui for pairs of vectors from the same Ui and
is zero on mixed pairs, i.e., κ(Ui, Uj) = 0 for i 6= j. Then κ is a Drinfeld orbifold algebra map
for G acting on
⊕r
i=1 Ui if and only if whenever i 6= j all group elements in the support of κ|Ui
act trivially on Uj.
Proof. For i = 1, . . . , r, suppose κ|Ui is a Drinfeld orbifold algebra map for G acting on Ui and
define κ on
∧2 (⊕r
i=1 Ui
)
as above. Conditions (2.1) and (2.2) of the definition of a Drinfeld
orbifold algebra map are straightforward to verify. For conditions (2.3)–(2.5), consider a triple of
vectors from
⊕r
i=1 Ui. If all three vectors are from the same Ui, then equations (2.3)–(2.5) hold
by virtue of κ|Ui being a Drinfeld orbifold algebra map. If the three vectors are from Ui, Uj ,
and Uk with i, j, k distinct, then conditions (2.3)–(2.5) are easily seen to hold because κ is
defined to be zero on pairs of vectors from different summands.
By multilinearity and skew-symmetry, all that remains is to examine the case where two
vectors, say u and v, are from the same Ui and the third vector, say w, is from some Uj with
j 6= i. Recall that
φ∗x,y(u, v, w) = κ∗x
(
u+ yu, κLy (v, w)
)
+ κ∗x
(
v + yv, κLy (w, u)
)
+ κ∗x
(
w + yw, κLy (u, v)
)
.
In the present case, the first two terms are zero because κLy (Ui, Uj) = 0, and the last term is zero
because κLy (Ui, Ui) ⊆ Ui and κ∗x(Uj , Ui) = 0. Thus φ∗x,y(u, v, w) = 0, which implies equation (2.5)
is satisfied for all g in G. We also see (2.3) and (2.4) will be satisfied if and only if for all g ∈ G,
we have ψ∗g(u, v, w) = 0 for ∗ = L and ∗ = C, respectively.
To this end, recall that
ψ∗g(u, v, w) = κ∗g(u, v)
(
gw − w
)
+ κ∗g(v, w)
(
gu− u
)
+ κ∗g(w, u)
(
gv − v
)
.
Continuing with u, v ∈ Ui and w ∈ Uj , we see that ψ∗g(u, v, w) = κ∗g(u, v)(gw − w) because
κ∗g(Ui, Uj) = 0. Thus for conditions (2.3) and (2.4) to hold for all g in G and all triples of this
type, it is both necessary and sufficient for group elements in the support of κ|Ui to act trivially
on Uj whenever i 6= j. �
As a corollary, in the case of two summands U1 and U2 with G acting trivially on U1 and
κ|U1 ≡ 0 (so that the support of κ|U1 is empty), we have:
Corollary 6.2. Let G be a finite group acting linearly on a vector space V = U1 ⊕ U2, where
each Ui is a subrepresentation. If G acts trivially on U1, then every Drinfeld orbifold algebra
map κ|U2 on
∧2 U2 extends to a Drinfeld orbifold algebra map κ on
∧2 V with imκL ⊆ U2 and
such that U1 ⊆ kerκ.
We now use Corollary 6.2 to show there are no Drinfeld orbifold algebra maps with linear
part supported only on the identity for Sn acting on h∗ ⊕ h.
Theorem 6.3. For Sn (n ≥ 3) acting on h∗⊕h ∼= C2n−2 by the doubled standard representation
there are no degree-one Lie orbifold algebra maps.
Proof. If there were a Lie orbifold algebra map κL+κC for the doubled standard representation
with κC :
∧2(h∗ ⊕ h)→ CSn and nonzero κL :
∧2(h∗ ⊕ h)→ h∗ ⊕ h, then it could be extended
as in Corollary 6.2 to yield a Lie orbifold algebra map for Sn acting on V = W ∗ ⊕W ∼= C2n
via the doubled permutation representation. The possible forms of Lie orbifold algebra maps κ
for the doubled permutation representation are controlled by Theorem 4.1, which includes the
PBW conditions γ1 = γ2 = γ4 = γ5 = 0 in (4.1). We will use these to show that in fact κL ≡ 0
Degree-One Rational Cherednik Algebras for the Symmetric Group 29
by first imposing the image constraint imκL ⊆ h∗ ⊕ h and the kernel constraint ι∗ ⊕ ι ⊆ kerκ
from Corollary 6.2.
First, use xi = x̄i+
1
nx[n] and yi = ȳi+
1
ny[n] in (3.3)–(3.6) to write the values of κL1 according
to the decomposition V ∼= h∗ ⊕ ι∗ ⊕ h⊕ ι:
κL1 (xi, xj) = a1(x̄i − x̄j) + b1(ȳi − ȳj),
κL1 (yi, yj) = a2(x̄i − x̄j) + b2(ȳi − ȳj),
κL1 (xi, yi) = a3x̄i +
1
n
(a3 + na4)x[n] + b3ȳi +
1
n
(b3 + nb4)y[n],
κL1 (xi, yj) = a5x̄i + a6x̄j +
1
n
(a5 + a6 + na7)x[n] + b5ȳi + b6ȳj +
1
n
(b5 + b6 + nb7)y[n].
Thus imκL ⊆ h∗ ⊕ h implies
a3 + na4 = 0, b3 + nb4 = 0,
a5 + a6 + na7 = 0, b5 + b6 + nb7 = 0.
Second, impose the extension conditions κL(wi, vj) = κL(vi, vj) = 0 for wi in the basis
{xi+1−xi, yi+1−yi | 1 ≤ i ≤ n−1} of h∗⊕h and vi, vj in the basis {x[n], y[n]} of ι∗⊕ι = (h∗⊕h)⊥.
From
κL(xi+1 − xi, x[n]) = na1(xi+1 − xi) + nb1(yi+1 − yi) = 0,
κL(yi+1 − yi, y[n] = na2(xi+1 − xi) + nb2(yi+1 − yi) = 0
we obtain a1 = b1 = a2 = b2 = 0. We also require that
κL(xi+1− xi, y[n]) = (a3+ na5− (a5+ a6))(xi+1− xi)+ (b3+ nb5− (b5+ b6))(yi+1− yi) = 0
and
κL(x[n], yi+1− yi) = (a3+ na6− (a5+ a6))(xi+1− xi)+ (b3+ nb6− (b5+ b6))(yi+1− yi) = 0,
and thus all four coefficients are zero. Using the results of the image constraint to simplify those
coefficients yields
a4 − a5 − a7 = 0, b4 − b5 − b7 = 0,
a4 − a6 − a7 = 0, b4 − b6 − b7 = 0,
from which it follows that a5 = a6 and b5 = b6. The remaining extension requirement imposes no
further constraints on the parameters because one verifies κL(x[n], y[n]) = 0 using a5+a6+na7 =
b5 + b6 + nb7 = 0.
To analyze the PBW conditions γ1 = γ2 = γ4 = γ5 = 0 in (4.1) it will help to first observe
that the above constraints a5 = a6 and a5 + a6 + na7 = 0 yield that
a5 = a6 = −n
2
a7,
and hence that
a4 = a5 + a7 = −n− 2
2
a7 and a3 = −na4 =
n(n− 2)
2
a7,
with corresponding expressions in terms of b7 for b3, b4, b5, and b6. These allow the simplification
φ1,1(xi, xj , yk) = [−b5a6+ b7(a3−a5−a6)](xi−xj)+ [b5(b5+ nb7)+ b7(b3− b5− b6)](yi−yj)
=
n2
4
a7b7(xi − xj) +
n2
4
b27(yi − yj).
30 B. Foster-Greenwood and C. Kriloff
Similarly,
φ1,1(yi, yj , xk) =
n2
4
a27(xi − xj) +
n2
4
a7b7(yi − yj).
Requiring each of these to be zero forces a7 = b7 = 0, and thus ai = bi = 0 for 3 ≤ i ≤ 6
as well. Since we already have a1 = b1 = a2 = b2 = 0, this proves there are no Lie orbifold
algebra maps for Sn acting on the doubled standard subrepresentation h∗ ⊕ h with κL 6≡ 0. �
For maps with linear part supported only off the identity there is instead a three-parameter
family of Drinfeld orbifold algebra maps that generalize the commutator relations for the rational
Cherednik algebra H0,c.
Theorem 6.4. Let Sn (n ≥ 3) act via the doubled standard representation on the space h∗⊕h ∼=
C2n−2 spanned by x̄i = xi + 1
nx[n] and ȳi = yi + 1
ny[n] with 1 ≤ i ≤ n. Let a⊥, b⊥, c ∈ C.
All Drinfeld orbifold algebra maps with nonzero linear part supported only off the identity have
the form κL + κC defined for 1 ≤ i 6= j ≤ n by
κL(x̄i, x̄j) = κL(ȳi, ȳj) = 0,
κL(x̄i, ȳi) = −n
2
∑
k 6=i
(
a⊥(x̄i + x̄k) + b⊥(ȳi + ȳk)
)
⊗ (ik),
κL(x̄i, ȳj) =
n
2
(
a⊥(x̄i + x̄j) + b⊥(ȳi + ȳj)
)
⊗ (ij)
and
κC(x̄i, x̄j) =
n2
4
(
b⊥
)2 ∑
k 6=i,j
(ijk)− (kji), κC(ȳi, ȳj) =
n2
4
(
a⊥
)2 ∑
k 6=i,j
(ijk)− (kji),
κC(x̄i, ȳi) = c
∑
k 6=i
(ik), κC(x̄i, ȳj) = −c(ij)− n2
4
a⊥b⊥
∑
k 6=i,j
(ijk)− (kji).
Proof. Suppose κ = κL + κC is a Drinfeld orbifold algebra map for the doubled standard
representation with κC :
∧2(h∗ ⊕ h) → CSn and nonzero κL :
∧2(h∗ ⊕ h) → (h∗ ⊕ h) ⊗ CSn
supported only off the identity. Extend κ as described in Corollary 6.2 to yield a Drinfeld orbifold
algebra map for Sn acting on V = W ∗ ⊕W ∼= C2n via the doubled permutation representation.
By Theorem 5.9 the possible forms of such extensions are κLref + κC1 + κCref + κC3-cyc satisfying the
PBW conditions (5.3) and (5.4).
We start by imposing the condition imκL ⊆ h∗ ⊕ h. Use x̄i = xi − 1
nx[n], ȳi = yi − 1
ny[n],
x̄ij := x̄i + x̄j , and ȳij := ȳi + ȳj to rewrite the nonzero values of κL(ij) as
κL(ij)(xi, yi) = κL(ij)(xj , yj) = −κL(ij)(xi, yj) = −κL(ij)(xj , yi)
= axij + a⊥x⊥ij + byij + b⊥y⊥ij =
(
a− a⊥
)
xij + a⊥x[n] +
(
b− b⊥
)
yij + b⊥y[n]
=
(
a− a⊥
)
x̄ij+
1
n
(
2a+ (n− 2)a⊥
)
x[n]+
(
b− b⊥
)
ȳij+
1
n
(
2b+ (n− 2)b⊥
)
y[n].
This shows 2a+ (n− 2)a⊥ = 2b+ (n− 2)b⊥ = 0 or
a− a⊥ = −n
2
a⊥ and b− b⊥ = −n
2
b⊥ (6.1)
and substituting these into (5.3) and (5.4) yields that
a(α− β) = a⊥(α− β) = b(α− β) = b⊥(α− β) = 0.
Thus since κL 6≡ 0 we must also have α = β in κC1 .
Degree-One Rational Cherednik Algebras for the Symmetric Group 31
Next consider conditions arising from ι∗ ⊕ ι ⊆ kerκ in Corollary 6.2, i.e., κC(w, v) =
κC(u, v) = 0 for all w in the basis {xi+1 − xi, yi+1 − yi | 1 ≤ i ≤ n − 1} of h∗ ⊕ h and u v
in the basis {x[n], y[n]} of ι∗ ⊕ ι = (h∗ ⊕ h)⊥.
For κC1 , since κC1 (xi, xj) = κC1 (yi, yj) = 0 and κC1 (xi, y[n]) = κC1 (x[n], yi) = α+(n−1)β for any
1 ≤ i, j ≤ n, the only extension condition that is not automatically satisfied is κC1 (x[n], y[n]) =
n(α+ (n− 1)β) = 0. Together with α = β this forces α = β = 0 and thus κC1 ≡ 0.
For κ∗ref and κC3-cyc, the definitions of κLg and κCg when g is a transposition and of κCg when g
is a 3-cycle yield that
κ∗g
(
v,
∑
h∈〈g〉
hv
)
= κ∗g
(
v,
∑
h∈〈g〉
hv∗
)
= 0 (6.2)
for all v ∈ {x1, . . . , xn, y1, . . . , yn}. This in turn implies that all of the extension conditions hold
for κ∗ref and κC3-cyc, yielding no further constraints on parameters.
We now evaluate κLref , κ
C
ref , and κC3-cyc at pairs of vectors from {x̄1, . . . , x̄n, ȳ1, . . . , ȳn}. For g
a transposition, i an index moved by g, v̄ a vector in {x̄i, gx̄i, ȳi, gȳi}, and ∗ = L or ∗ = C,
we have κ∗g(v̄,
gv̄) = 0 and we use (6.2) to observe that for 1 ≤ i ≤ n,
κ∗g(x̄i, ȳi) = −κ∗g(x̄i, gȳi) = κ∗g(xi, yi)−
1
n
κ∗g(xi, y[n])−
1
n
κ∗g(x[n], yj) +
1
n2
κ∗g(x[n], y[n])
= κ∗g(xi, yi).
It then follows from (6.1) that
κLg (x̄i, ȳi) = −κLg (x̄i,
gȳi) = −n
2
a⊥x̄ij −
n
2
b⊥ȳij ,
κCg (x̄i, ȳi) = −κCg (x̄i,
gȳi) = c.
For g a 3-cycle, by the orbit property in (6.2) and by (6.1) we see that
κCg (v̄, v̄∗) = 0,
κCg (v̄, gv̄) =
n2
4
(
a⊥
)2
if v ∈W,
n2
4
(
b⊥
)2
if v ∈W ∗,
κCg (gv̄, v̄∗) = −κCg (v̄, gv̄∗) =
n2
4
a⊥b⊥.
These components produce the given definition of κL + κC . �
Remark 6.5. In the case κL1 = κLref ≡ 0 then also κC3-cyc ≡ 0, α = −(n−1)β, and the restriction
of the constant 2-cochain κC = κC1 + κCref to
∧2(h∗ ⊕ h) is given by
κC1 (x̄i, x̄j) = κC1 (ȳi, ȳj) = 0, κC1 (x̄i, ȳi) = −(n− 1)β, κC1 (x̄i, ȳj) = β,
κCg (x̄i, x̄j) = κCg (ȳi, ȳj) = 0, κCg (x̄i, ȳi) = c, κCg (x̄i, ȳj) = −c,
where β, c ∈ C, g is a transposition, and 1 ≤ i 6= j ≤ n. This corresponds to the rational
Cherednik algebra Hnβ,c.
In the theory of rational Cherednik algebras, Ht,c, for the symmetric group, a natural iso-
morphism between Ht,c and Hλt,λc when λ ∈ C× means that only two distinct cases need be
considered, t 6= 0 and t = 0. Theorems 6.3 and 6.4 show that in the first case there are no further
deformations in polynomial degree one with the linear part of the parameter map supported only
on the identity while there is a three-parameter family of such deformations in the second case
with the linear part of the parameter map supported only off the identity.
32 B. Foster-Greenwood and C. Kriloff
Remark 6.6. What if κL were supported both on and off the identity? The specializations
of parameter values for κL1 in part (1) and for κLref in part (3) of Theorem 5.10 on the combined
lift of κL1 + κLref means that
κL1 (xi, yi) = κL1 (xi, yj) = a4x[n] + b4y[n],
κL(ij)(xi, yi) = −κL(ij)(xi, yj) = −n
2
a⊥(x̄i + x̄j)−
n
2
b⊥(ȳi + ȳj).
But then im
(
κL1 + κLref
)
⊆ h∗ ⊕ h would require a4 = b4 = 0 so κL1 ≡ 0. This combined with
part (2) of Theorem 5.10 shows there is no Drinfeld orbifold algebra map for Sn on h∗ ⊕ h with
linear part supported both on and off the identity which extends to a map κ of the form in
Theorem 5.10. But since Theorem 5.10 is not exhaustive, it is not clear whether there exist such
maps in general.
7 Descriptions of degree-one rational Cherednik algebras
Here we present, via generators and relations, degree-one PBW deformations of the skew group
algebras S(W ∗ ⊕W )#Sn and S(h∗ ⊕ h)#Sn that result from Theorems 4.1, 5.9, and 6.4 when
n ≥ 3. This facilitates comparison with degree-zero deformations (i.e., rational Cherednik alge-
bras) and with the PBW deformations of S(W )#Sn in [8]. The classifications are summarized
in Tables 2 and 3. We reiterate that the case when n = 2 can be analyzed in similar fashion,
but involves some differences in the dimensions of spaces of pre-Drinfeld orbifold algebra maps
and in the parameter relations required in order to satisfy the PBW conditions.
7.1 Algebras for the doubled permutation representation
First, the Lie orbifold algebra maps involving 17 parameters classified in Theorem 4.1 yield
a variety controlling the Lie orbifold algebras that deform S(W ∗⊕W )#Sn in degree one. Based
on representative calculations in Macaulay2 [14] we conjecture that this projective variety is
of dimension seven. Some subvarieties of potential interest are indicated in Table 1 in Section 4.
When κL1 ≡ 0 these Lie orbifold algebras specialize to rational Cherednik algebras corresponding
to the parameter c and the general G-invariant skew-symmetric bilinear form κC1 involving α
and β (because W is decomposable — see [6, proof of Theorem 1.3]).
Theorem 7.1 (Lie orbifold algebras for doubled permutation representation over C[t]). Let Sn
(n ≥ 3) act on V = W ∗ ⊕W with basis B = {x1, . . . , xn, y1, . . . , yn} by the doubled permutation
representation. For a1, . . . , a7, b1, . . . , b7, α, β, c ∈ C subject to conditions (4.1), (4.2), and (4.3),
define κL = κL1 and κC = κC1 + κCref to be the linear and constant cochains such that for 1 ≤ i 6=
j ≤ n,
κL(xi, xj) = (a1(xi − xj) + b1(yi − yj)), κC(xi, xj) = 0,
κL(yi, yj) = (a2(xi − xj) + b2(yi − yj)), κC(yi, yj) = 0,
κL(xi, yi) = (a3xi + a4x[n] + b3yi + b4y[n]), κC(xi, yi) = α+ c
∑
k 6=i
(ik),
κL(xi, yj) = (a5xi + a6xj + a7x[n] + b5yi + b6yj + b7y[n]), κC(xi, yj) = β − c(ij).
Then the quotient Hκ,t of T (V )#Sn[t] by the ideal generated by{
uv − vu− κL(u, v)t− κC(u, v)t2 | u, v ∈ B
}
is a Lie orbifold algebra over C[t]. In fact, the algebras Hκ,1 are precisely the Drinfeld orbifold
algebras such that κL is supported only on the identity.
Degree-One Rational Cherednik Algebras for the Symmetric Group 33
Table 2. Classification of Drinfeld orbifold algebra maps for Sn acting on W ∗ ⊕W .
Linear part κL Constant part κC Parameter relations Reference
κL1 0 (4.1) Theorem 4.1
κC1 (4.1)–(4.2)
κC1 + κCref with κCref 6≡ 0 (4.4)–(4.17)
κLref κC3-cyc none Theorem 5.9
κC3-cyc + κCref none
κC3-cyc + κCref + κC1 (5.3)–(5.4)
κL1 + κLref κC3-cyc + κCref + κC1 Theorem 5.10(1)–(3) Theorem 5.10
? ?
0 κC1 + κCref none
When the indicated parameter relations are satisfied, the map κ = κL +κC is a Drinfeld orbifold algebra
map. The question marks indicate there could be further maps with κL = κL1 + κLref .
Second, for κL supported only off the identity, Theorem 5.9 shows that by comparison there
is only a seven-parameter family of Drinfeld orbifold algebra maps and these are controlled by
a projective variety which, according to a few representative calculations in Macaulay2 [14],
appears to be four-dimensional. The resulting algebras also specialize to rational Cherednik
algebras parametrized by α, β, and c when κLref = κC3-cyc ≡ 0.
Theorem 7.2 (Drinfeld orbifold algebras for doubled permutation representation over C[t]).
Let Sn (n ≥ 3) act on V = W ∗ ⊕ W with basis B = {x1, . . . , xn, y1, . . . , yn} by the doubled
permutation representation. Suppose a, a⊥, b, b⊥, c, α, β ∈ C satisfy conditions (5.3) and (5.4).
Define κL = κL1 and κC = κC1 + κCref + κC3-cyc to be the cochains such that for 1 ≤ i 6= j ≤ n,
κL(xi, xj) = κL(yi, yj) = 0,
κL(xi, yi) =
∑
k 6=i
((
a− a⊥
)
xi,k + a⊥x[n] +
(
b− b⊥
)
yi,k + b⊥y[n]
)
⊗ (ik),
κL(xi, yj) = −
((
a− a⊥
)
xi,j + a⊥x[n] +
(
b− b⊥
)
yi,j + b⊥y[n]
)
⊗ (ij)
and
κC(xi, yj) = β − c(ij)−
(
a− a⊥
)(
b− b⊥
) ∑
k 6=i,j
(ijk)− (kji),
κC(xi, xj) =
(
b− b⊥
)2 ∑
k 6=i,j
(ijk)− (kji),
κC(yi, yj) =
(
a− a⊥
)2 ∑
k 6=i,j
(ijk)− (kji),
κC(xi, yi) = α+ c
∑
k 6=i
(ik).
Then the quotient Hκ,t of T (V )#Sn[t] by the ideal generated by{
uv − vu− κL(u, v)t− κC(u, v)t2 | u, v ∈ B
}
is a Drinfeld orbifold algebra over C[t]. Further, the algebras Hκ,1 are precisely the Drinfeld
orbifold algebras such that imκLg ⊆ V g for each g ∈ Sn and κL is supported only off the identity.
An analogous statement may be made for algebras constructed from the family of lifts of
κL1 + κLref described in Theorem 5.10 but is omitted here.
34 B. Foster-Greenwood and C. Kriloff
Table 3. Classification of Drinfeld orbifold algebra maps for Sn acting on h∗ ⊕ h.
Linear part κL Constant part κC Parameter relations Reference
κLref κC3-cyc 2a+ (n− 2)a⊥ = 0 Theorem 6.4
2b+ (n− 2)b⊥ = 0
κC3-cyc + κCref 2a+ (n− 2)a⊥ = 0
2b+ (n− 2)b⊥ = 0
0 κCref none Remark 6.5
κC1 α+ (n− 1)β = 0
κC1 + κCref α+ (n− 1)β = 0
When the parameter relations hold, the map κ = κL + κC is a Drinfeld orbifold algebra map.
7.2 Algebras for the doubled standard representation
By Theorem 6.3 the only Lie orbifold algebras for Sn acting on h∗ ⊕ h by the doubled standard
representation are the known rational Cherednik algebras Hnβ,c. However, by Theorem 6.4 there
is in this case a three-parameter family of Drinfeld orbifold algebras which are not graded Hecke
algebras, but which specialize when a⊥ = b⊥ = 0 to the rational Cherednik algebras H0,c for Sn.
Theorem 7.3 (Drinfeld orbifold algebras for doubled standard representation over C[t]). Let Sn
(n ≥ 3) act on h∗ ⊕ h by the doubled standard representation. For a⊥, b⊥, c ∈ C and x̄i and ȳi
as in (3.1) define κL and κC as in Theorem 6.4. Then the quotient Hκ,t of T (h∗ ⊕ h)#Sn[t] by
the ideal generated by{
ūv̄ − v̄ū− κL(ū, v̄)t− κC(ū, v̄)t2 | ū, v̄ ∈ B̄
}
is a Drinfeld orbifold algebra of S(h∗⊕h)#Sn over C[t]. Further, the algebras Hκ,1 are precisely
the Drinfeld orbifold algebras such that imκLg ⊆ (h∗ ⊕ h)g for each g ∈ Sn and κL is supported
only off the identity. Specializing a⊥ = b⊥ = 0 yields the rational Cherednik algebra H0,c.
Acknowledgements
We thank the referees for helpful suggestions and questions that improved the writing of the
paper, particularly those that prompted us to add Proposition 6.1, improve Section 4.5, and
reorganize overall. We also thank Lily Silverstein for helpful conversations related to Section 4.5.
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1 Introduction
2 Preliminaries
2.1 Skew group algebras
2.2 Cochains
2.3 Drinfeld orbifold algebras
2.4 Lie orbifold algebras
2.5 Drinfeld orbifold algebra maps
2.6 Drinfeld orbifold algebra maps (condensed definition)
2.7 Strategy
2.8 Hochschild cohomology to pre-DOA maps
3 Pre-Drinfeld orbifold algebra maps
3.1 Pre-Drinfeld orbifold algebra maps supported only on the identity
3.2 Pre-Drinfeld orbifold algebra maps supported only off the identity
3.3 Pre-Drinfeld orbifold algebra maps
3.4 Definitions of linear and constant cochains
4 Lie orbifold algebra maps that deform S(W* + W)Sn
4.1 All basis vectors in W or in W* and three distinct indices
4.2 Two basis vectors in W or W* and three distinct indices
4.3 Two basis vectors in W or W* and two distinct indices
4.4 Lie orbifold algebra maps
4.5 Algebraic varieties corresponding to image constraints
5 Drinfeld orbifold algebra maps that deform S(W* + W) Sn
5.1 Clearing obstructions to deformations of S(W* + W) Sn
5.2 Clearing the first obstruction
5.3 Clearing the second obstruction
5.4 Drinfeld orbifold algebra maps
6 Drinfeld orbifold algebra maps that deform S(h* + h)Sn
7 Descriptions of degree-one rational Cherednik algebras
7.1 Algebras for the doubled permutation representation
7.2 Algebras for the doubled standard representation
References
|
| id | nasplib_isofts_kiev_ua-123456789-211310 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T07:33:23Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Foster-Greenwood, Briana Kriloff, Cathy 2025-12-29T11:08:22Z 2021 Degree-One Rational Cherednik Algebras for the Symmetric Group. Briana Foster-Greenwood and Cathy Kriloff. SIGMA 17 (2021), 039, 35 pages 1815-0659 2020 Mathematics Subject Classification: 16S80; 16E40; 16S35; 20B30 arXiv:1912.06743 https://nasplib.isofts.kiev.ua/handle/123456789/211310 https://doi.org/10.3842/SIGMA.2021.039 Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized families of Drinfeld orbifold algebras for symmetric groups acting on doubled representations that generalize rational Cherednik algebras by deforming in degree one. We characterize rich families of maps recording commutator relations with their linear parts supported only on and only off the identity when the symmetric group acts on the natural permutation representation plus its dual. This produces degree-one versions of 𝖌𝔩ₙ-type rational Cherednik algebras. When the symmetric group acts on the standard irreducible reflection representation plus its dual, there are no degree-one Lie orbifold algebra maps, but there is a three-parameter family of Drinfeld orbifold algebras arising from maps supported only off the identity. These provide degree-one generalizations of the 𝔰𝔩ₙ-type rational Cherednik algebras 𝐻₀ ̦c. We thank the referees for helpful suggestions and questions that improved the writing of the paper, particularly those that prompted us to add Proposition6.1, improve Section 4.5, and reorganize overall. We also thank Lily Silverstein for helpful conversations related to Section 4.5. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Degree-One Rational Cherednik Algebras for the Symmetric Group Article published earlier |
| spellingShingle | Degree-One Rational Cherednik Algebras for the Symmetric Group Foster-Greenwood, Briana Kriloff, Cathy |
| title | Degree-One Rational Cherednik Algebras for the Symmetric Group |
| title_full | Degree-One Rational Cherednik Algebras for the Symmetric Group |
| title_fullStr | Degree-One Rational Cherednik Algebras for the Symmetric Group |
| title_full_unstemmed | Degree-One Rational Cherednik Algebras for the Symmetric Group |
| title_short | Degree-One Rational Cherednik Algebras for the Symmetric Group |
| title_sort | degree-one rational cherednik algebras for the symmetric group |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211310 |
| work_keys_str_mv | AT fostergreenwoodbriana degreeonerationalcherednikalgebrasforthesymmetricgroup AT kriloffcathy degreeonerationalcherednikalgebrasforthesymmetricgroup |