Mystic Reflection Groups
This paper aims to systematically study mystic reflection groups that emerged independently in the paper [Selecta Math. (N.S.) 14 (2009), 325-372] by the authors and in the paper [Algebr. Represent. Theory 13 (2010), 127-158] by Kirkman, Kuzmanovich and Zhang. A detailed analysis of this class of gr...
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Bazlov, Y. Berenstein, A. 2019-02-11T16:21:38Z 2019-02-11T16:21:38Z 2014 Mystic Reflection Groups / Y. Bazlov, A. Berenstein // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 2 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16G99; 20F55; 16S80 DOI:10.3842/SIGMA.2014.040 https://nasplib.isofts.kiev.ua/handle/123456789/146818 This paper aims to systematically study mystic reflection groups that emerged independently in the paper [Selecta Math. (N.S.) 14 (2009), 325-372] by the authors and in the paper [Algebr. Represent. Theory 13 (2010), 127-158] by Kirkman, Kuzmanovich and Zhang. A detailed analysis of this class of groups reveals that they are in a nontrivial correspondence with the complex reflection groups G(m,p,n). We also prove that the group algebras of corresponding groups are isomorphic and classify all such groups up to isomorphism. This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html. We thank Ken Brown for bringing the paper [2] to our attention, and Alexander Premet for stimulating discussions. The present paper was started when both authors were research members of the Mathematical Sciences Research Institute. We thank the Institute and the organizers of the Noncommutative Algebraic Geometry and Representation Theory program for creating an atmosphere conducive for research. We acknowledge partial support of the LMS Research in Pairs grant ref. 41224. The second named author was partially supported by the NSF grant DMS-1101507. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Mystic Reflection Groups Article published earlier |
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This paper aims to systematically study mystic reflection groups that emerged independently in the paper [Selecta Math. (N.S.) 14 (2009), 325-372] by the authors and in the paper [Algebr. Represent. Theory 13 (2010), 127-158] by Kirkman, Kuzmanovich and Zhang. A detailed analysis of this class of groups reveals that they are in a nontrivial correspondence with the complex reflection groups G(m,p,n). We also prove that the group algebras of corresponding groups are isomorphic and classify all such groups up to isomorphism.
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Mystic Reflection Groups / Y. Bazlov, A. Berenstein // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 2 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 040, 11 pages
Mystic Reflection Groups?
Yuri BAZLOV † and Arkady BERENSTEIN ‡
† School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK
E-mail: yuri.bazlov@manchester.ac.uk
‡ Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
E-mail: arkadiy@math.uoregon.edu
Received December 25, 2013, in final form March 24, 2014; Published online April 04, 2014
http://dx.doi.org/10.3842/SIGMA.2014.040
Abstract. This paper aims to systematically study mystic reflection groups that emerged
independently in the paper [Selecta Math. (N.S.) 14 (2009), 325–372] by the authors and
in the paper [Algebr. Represent. Theory 13 (2010), 127–158] by Kirkman, Kuzmanovich
and Zhang. A detailed analysis of this class of groups reveals that they are in a nontrivial
correspondence with the complex reflection groups G(m, p, n). We also prove that the
group algebras of corresponding groups are isomorphic and classify all such groups up to
isomorphism.
Key words: complex reflection; mystic reflection group; thick subgroups
2010 Mathematics Subject Classification: 16G99; 20F55; 16S80
1 Introduction
Let V be a complex vector space with basis {x1, . . . , xn}. Denote by S(V ) the symmetric algebra
of V . It is a fundamental fact that the algebra S(V )Sn of symmetric polynomials is isomorphic
to S(V ). More generally, the Chevalley–Shephard–Todd theorem asserts that for a finite group
G ⊂ GL(V ), S(V )G is isomorphic to S(V ) if and only if G is generated by complex reflections
on V .
In a remarkable paper [2], Kirkman, Kuzmanovich and Zhang solved the following problem:
Problem 1.1. Given a complex matrix q = (qij , 1 ≤ i, j ≤ n) with qijqji = 1, qii = 1, let Sq(V )
be the algebra generated by V subject to the relations xixj = qijxjxi for 1 ≤ i, j ≤ n. Find all
finite groups G such that
(a) G acts on the algebra Sq(V ) by degree-preserving automorphisms;
(b) The fixed point algebra Sq(V )G is isomorphic to Sq′(V ) for some q′.
We will refer to a group G satisfying (a) and (b) above as a mystic reflection group.
Independently, in [1] we solved the following problem:
Problem 1.2. Classify all algebras A such that
• A is generated by the space V , its dual V ∗ and a finite group G ⊂ GL(V );
• A admits a triangular decomposition A = Sq(V )⊗ CG⊗ Sq(V ∗) for some q as above;
• Sq(V ) and Sq(V ∗) are invariant under conjugation by elements of G, and the restriction of
these conjugation representations to V and V ∗ is the natural action of G on these spaces;
• yjxi− qijxiyj ∈ CG for all i, j, where {y1, . . . , yn} is the basis of V ∗ dual to {x1, . . . , xn}.
?This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full
collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html
mailto:yuri.bazlov@manchester.ac.uk
mailto:arkadiy@math.uoregon.edu
http://dx.doi.org/10.3842/SIGMA.2014.040
http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html
2 Y. Bazlov and A. Berenstein
Comparing [2, Theorem 1.1 and Corollary 5.6] and [1, Theorem 0.7], one obtains the following
surprising result:
Theorem 1.3. A group G solves Problem 1.2 if and only if it is a mystic reflection group.
The goal of this paper is to “demystify” the mystic reflection groups, develop their structural
theory, show that their group algebras are isomorphic to those of complex reflection groups, and
to deduce Problem 1.1 directly from the classical Chevalley–Shepard–Todd theorem.
2 Main results
We start with a new notion of “mystical equivalence” of group actions, which is crucial for what
follows.
Definition 2.1. Let . : G × V → V, .′ : G′ × V → V be faithful actions of finite groups G,
respectively G′, on a complex vector space V. We say that the actions . and .′ are mystically
equivalent, if
ρ(eG) = ρ′(eG′),
where ρ : CG→ EndC V and ρ′ : CG′ → EndC V are the algebra homomorphisms defined by the
actions, and eG denotes the element
∑
g∈G
g of CG.
Mystical equivalence of the actions of G and G′ is a strengthening of the condition that the
respective spaces VG and VG′ of invariants are equal, due to the following obvious result.
Lemma 2.2. If a G-action and a G′-action on V are mystically equivalent, then VG = VG′.
We will use the lemma in the situation where V = S(V ) where V is a vector space over C
with a chosen basis {x1, . . . , xn}. Throughout the paper, n ≥ 2. Denote by Gn the group of
monomial matrices on V , that is, matrices in GLn(C) with exactly n non-zero entries. In other
words,
Gn = (C×)n o Sn.
Here (C×)n is naturally identified with the group of diagonal matrices in GLn(C) and acts
on V by scaling the basis {x1, . . . , xn}. The symmetric group Sn is identified with the group
of permutation matrices and acts on V by permuting the same basis. Note that Gn is the
normalizer of the torus (C×)n in GLn(C). In particular, Sn acts on (C×)n by conjugation.
When we write tw ∈ Gn, we will imply that t ∈ (C×)n and w ∈ Sn; every element of Gn can be
uniquely written in this way.
Clearly, Gn is generated by s1, . . . , sn−1 and t
(ζ)
j , 1 ≤ j ≤ n, ζ ∈ C×, where
• si ∈ Sn is the permutation of {x1, . . . , xn} which swaps xi and xi+1;
• t(ζ)j ∈ (C×)n maps xk to ζδjkxk.
We find it very convenient to use the linear character
det : Gn → C×
of Gn, which is just the restriction of the determinant character of GLn(C) to Gn. In particular,
det si = −1, detw ∈ {±1} for w ∈ Sn, det
(
t
(ζ1)
1 t
(ζ2)
2 · · · t(ζn)n
)
= ζ1ζ2 · · · ζn.
Next, we introduce two different faithful actions of Gn on S(V ) using the natural basis{
xk11 · · ·xknn : k1, . . . , kn ∈ Z≥0
}
of S(V ). (At the moment, we are not using any multiplication
on S(V ).)
Mystic Reflection Groups 3
Proposition 2.3.
(a) There exist (unique) faithful actions .+, .− of Gn on S(V ) such that
si .± x
k1
1 · · ·x
kn
n = (±1)kiki+1xk11 · · ·x
ki+1
i xkii+1 · · ·x
kn
n ,
t
(ζ)
j .± x
k1
1 · · ·x
kn
n = ζkjxk11 · · ·x
kn
n
for any i = 1, . . . , n − 1, j = 1, . . . , n, ζ ∈ C×. Both actions extend the defining action
of Gn on V .
(b) The action .+ of Gn on S(V ) is compatible with the natural commutative multiplication
on S(V ), in the sense that Gn acts by automorphisms of the algebra S(V ).
(c) The action .− of Gn on S(V ) is compatible with the algebra structure S−1(V ) on S(V )
(which is Sq(V ) with qij = −1 for all i 6= j);
(d) ρ+(CGn) = ρ−(CGn), where ρ± : CGn → EndC S(V ) are the algebra homomorphisms
arising from the actions .±. Moreover, ρ− = ρ+ ◦ J for some algebra automorphism J
of CGn.
Remark 2.4. By a powerful result on group actions on integral domains, see Corollary 4.2
from the Appendix taken with R = C, A = S(V ), the action .+ is faithful, and moreover the
corresponding algebra homomorphism ρ+ : CGn → EndC S(V ) is injective. It follows from (d)
that ρ− : CGn → EndC S−1(V ) is also injective. This fact does not readily follow from classical
results.
At this point, we restrict our attention to finite subgroups G of Gn – namely, to Shephard–
Todd’s imprimitive complex reflection groups G(m, p, n) and the groups WC,C′ , introduced in-
dependently in [1] and [2] and defined as follows. Let n ≥ 1 and C′ ⊆ C be two finite subgroups
of C× of orders m
p , m respectively. Then
G(m, p, n) = {tw ∈ Cn o Sn : det t ∈ C′},
WC,C′ = {tw ∈ Cn o Sn : det(tw) ∈ C′}.
The similarity of the two definitions manifests itself in our first main result where a correspon-
dence, µ, between the two classes of subgroups of Gn is established.
Main Theorem 2.5. Given n ≥ 1, an even m ≥ 2 and a divisor p of m, let G = G(m, p, n).
Then there exists a unique finite subgroup µ(G) ⊂ Gn such that the restriction of .− onto µ(G)
is mystically equivalent to .+ on G. In fact,
µ(G) = WC,C′ ,
where |C| = m, |C′| = m
p .
The definition of the group µ(G) suggests that the invariants of µ(G(m, p, n)) = WC,C′ should
be viewed in the noncommutative algebra S−1(V ), where this group acts via .−. To describe
these invariants, introduce the following elements in the space S(V ):
p
(m)
k =
n∑
i=1
xkmi , k = 1, . . . , n− 1, r(l) = xl1x
l
2 · · ·xln.
The classical result of Shephard–Todd and Chevalley asserts that the subalgebra S(V )G(m,p,n)
of S(V ) (with respect to the natural commutative product on S(V )) equals the polynomial
algebra C
[
p
(m)
1 , . . . , p
(m)
n−1, r
(m
p
)]
.
Our mystic equivalence construction immediately leads to the following result, first obtained
by Kirkman, Kuzmanovich and Zhang as a key ingredient in the classification theorem [2,
Theorem 1.1].
4 Y. Bazlov and A. Berenstein
Theorem 2.6. In the notation of Theorem 2.5, let m be even. Then in the algebra S−1(V ),
(a) the elements p
(|C|)
1 , . . . , p
(|C|)
n−1, r
(|C′|) are pairwise commuting invariants of the group WC,C′;
(b) S−1(V )WC,C′ = C
[
p
(|C|)
1 , . . . , p
(|C|)
n−1, r
(|C′|)].
Remark 2.7. This result shows that the condition that m is even in Theorem 2.5 is important:
the symmetric group Sn = G(1, 1, n) does not have a mystical counterpart, and indeed the
correspondence µ cannot be extended to groups G(m, p, n) where m is odd. This happens
because the space of the invariants of G(m, p, n) in S(V ) is not closed under the multiplication
in S−1(V ), and therefore cannot be the space of invariants of a group acting by automorphisms
of S−1(V ).
The groups G = G(m, p, n) and µ(G) are of the same order mnn!
p , and, informally, they “look
very similar”. Our next main result makes this informal statement more precise. We keep the
notation from Theorem 2.5 and denote by R the ring Z
[
1+i
2
]
⊂ C.
Main Theorem 2.8. For all G as in Theorem 2.5, the group rings RG and Rµ(G) are isomor-
phic. In particular, CG ∼= Cµ(G).
This theorem is rather nontrivial because the groups G and µ(G) are often not isomorphic
as abstract groups. Indeed, it was shown in [2, Example 7.3] that the group G = G(2, 2, n) is
not isomorphic to its mystic counterpart µ(G) = W{±1},{1} for all even n. We generalize this
observation and give a complete list of cases where G is not isomorphic to µ(G).
Theorem 2.9. In the notation of Theorem 2.5, let G = G(m, p, n) with m even. Then the
groups G and µ(G) are not isomorphic as abstract groups, if and only if n is even and m
p is odd.
We go further than this and classify all groups of the form G = G(m, p, n) and µ(G) up to
isomorphism. We need the following useful notion.
Definition 2.10. We say that a subgroup G of a semidirect product T oH is thick if
• π(G) = H, where π : T oH → H is the canonical projection onto the second factor;
• G is normal in in T oH.
It is not difficult to see that all the groups from Theorem 2.5 are thick subgroups of
G(m, 1, n) = Cn o Sn. It turns out that a converse is also true.
Theorem 2.11. Let C be the subgroup of C× of order m. Then every thick subgroup of CnoSn
is of the form G(m, p, n) or (if m is even) WC,C′, and in particular, is a mystic reflection group.
The following completes the classification of thick subgroups of all G(m, 1, n) up to isomor-
phism. We keep the notation used in the preceding theorems.
Theorem 2.12. Let G ⊆ G(m, 1, n), G′ ⊆ G(m′, 1, n′) be thick subgroups.
(a) Suppose that n = n′ and G 6= G′ in Gn. Then G ∼= G′ if and only if n is odd, m = m′ is
even, and {G,G′} = {G(m, p, n), µ(G(m, p, n))} for some p such that m
p is odd.
(b) Suppose that n<n′. Then G ∼= G′ if and only if n=3, n′=4, G∈{G(2, 2, 3), µ(G(2, 2, 3))},
G′ = G(1, 1, 4) = S4.
Remark 2.13. It is not difficult to see that if m
p is even (in the notation of Theorem 2.5), then
G = µ(G) in Gn. This theorem together with Theorem 2.8 implies that the converse is also true.
The above classification suggests the following general problem which we do not address in
the present paper:
Problem 2.14. Given a semidirect product group T oH where T and H are finite, classify all
finite thick subgroups of T oH up to isomorphism.
Mystic Reflection Groups 5
3 Proofs of results from Section 2
We will repeatedly use the following straightforward technical fact about the root system of
type An−1, the proof of which is left to the reader as an exercise.
Claim 3.1. Let A be a multiplicatively written abelian group. Let φij : Zn → A, 1 ≤ i, j ≤ n,
i 6= j, be a system of maps satisfying
φij(w(k)) = φw−1(i),w−1(j)(k), φji(k)φij(k) = 1
for all k = (k1, . . . , kn) ∈ Zn, w ∈ Sn, where w(k) stands for (kw−1(1), . . . , kw−1(n)). Denote
φw(k) =
∏
i<j,w(i)>w(j)
φij(k).
Then:
(a) For all w′, w ∈ Sn,
φw′w(k) = φw′(w(k))φw(k).
(b) If for all i 6= j and for all k,k′ ∈ Zn one has
φij(k + k′) = a
kik
′
j−k′ikj
ij φij(k)φij(k
′),
then for all w ∈ Sn one has
〈k,k′〉φw(k + k′) = 〈w(k), w(k′)〉φw(k)φw(k′),
where 〈k,k′〉 =
∏
i<j
a
k′ikj
ij . Here aij are elements of A such that aji = aij.
Proof of Proposition 2.3
Observe that the group Gn ⊂ GLn(C) is generated by Sn and (C×)n subject to the semidirect
product relations
wt
(ζ)
j = t
(ζ)
w(j)w, for all w ∈ Sn, j ∈ {1, . . . , n}, ζ ∈ C×.
In what follows, we use the abbreviation xk to denote xk11 · · ·xknn , where k = (k1, . . . , kn) ∈ Zn≥0.
For c ∈ C×, define φ
(c)
ij : Zn → C× by
φ
(c)
ij (k) = (−1)kikjcki−kj , where k = (k mod 2) ∈ {0, 1}.
Additionally, define φ
(0)
ij (k) = 1 for all k. Because φ
(c−1)
ij (k) = φ
(c)
ij (k)−1 = φ
(c)
ji (k), the sys-
tem φ
(c)
ij satisfies the condition in Claim 3.1 and hence gives rise to functions φ
(c)
w . The following
lemma is then immediate from Claim 3.1(a).
Lemma 3.2. For each c ∈ C, the formula
wt
(ζ1)
1 · · · t(ζn)n .c x
k = φ(c)w (k)
n∏
j=1
ζ
kj
j
xw(k)
defines an action .c of the group Gn on the space S(V ).
6 Y. Bazlov and A. Berenstein
The actions .0, .1 given by Lemma 3.2 coincide on the generators si, t
(ζ)
j of Gn with the
actions .+, .− defined in part (a) of Proposition 2.3. Hence part (a) of the proposition is
proved.
Denote by •+, respectively •−, the multiplication on the algebra S(V ), respectively S−1(V ).
One has the following multiplication rule for monomials:
xk •± xk
′
= 〈k,k′〉±xk+k′ for all k,k′ ∈ Zn≥0,
where 〈k,k′〉± is as given in Claim 3.1(b) with all aij = ±1, i 6= j.
It is enough to check that w .± and t
(ζ)
j .± are automorphisms of the respective algebra
structures on S(V ). We apply these actions to both sides of the multiplication rule for monomials
and check that the results are equal. This is trivial for t
(ζ)
j .±. For w .± where w ∈ Sn,
the equality is guaranteed by Claim 3.1(b) and Lemma 3.2, applied to functions φ+ij = φ
(0)
ij ,
respectively φ−ij = φ
(1)
ij . This proves parts (b), (c) of Proposition 2.3.
Now let us prove part (d) of Proposition 2.3. Clearly, the natural Sn-action on the group
(C×)n ⊂ Gn extends to that on the group algebra C(C×)n.
For each c ∈ C \ {0}, w ∈ Sn, define the element Q
(c)
w ∈ C(C×)n by
Q(c)
w =
∏
i<j:w(i)>w(j)
Q
(c)
ij ,
where Q
(c)
ij = 1
4
((
c+ c−1
)(
1− t(−1)i t
(−1)
j
)
+
(
c− c−1 + 2
)
t
(−1)
i +
(
c−1 − c+ 2
)
t
(−1)
j
)
, 1 ≤ i, j ≤ n.
Lemma 3.3. For all w′, w ∈ Sn, c ∈ C×, Q
(c)
w Q
(c−1)
w = 1 and Q
(c)
w′w = w−1(Q
(c)
w′ ) ·Q
(c)
w .
Proof. Denote by F the algebra of all functions from Zn to C with pointwise addition and
multiplication. Clearly, the assignment t
(ζ1)
1 . . . t
(ζn)
n 7→ (k 7→ ζk11 . . . ζknn ) defines a group homo-
morphism (C×)n → F× which extends to an injective map Ψ: C(C×)n ↪→ F .
It is easy to check that Ψ(Q
(c)
ij ) = φ
(c)
ij where φ
(c)
ij is as in Lemma 3.2. In particular,
Q
(c)
ij Q
(c−1)
ij = 1 since φ
(c)
ij φ
(c−1)
ij = 1 and Ψ is injective. This proves the first assertion of the
lemma. To prove the second assertion, apply Ψ−1 to Claim 3.1(a) and use the fact that the
function k 7→ φ(w(k)) is mapped by Ψ−1 to w−1(Ψ−1(φ)) for all functions φ from the subgroup
of F× generated by {φ(c)ij : i 6= j}. �
Now for each c ∈ C× define the C-linear map Jc : CGn → CGn by the formula
Jc(wt) = wtQ(c)
w , t ∈ C(C×)n, w ∈ Sn.
Lemma 3.4. For each n ≥ 1,
(a) Jc is an algebra automorphism of CGn with inverse Jc−1.
(b) ρ+ ◦ Jc = ρc, where ρc : CGn → EndC S(V ) is the algebra homomorphism corresponding
to .c.
Proof. On the one hand, Jc(w
′t′wt) = Jc(w
′w · w−1(t′)t) = w′w · w−1(t′)tQ(c)
w′w. On the other
hand,
Jc(w
′t′)Jc(wt) = w′t′Q
(c)
w′wtQ
(c)
w = w′w · w−1(t′)tw−1(Q(c)
w′ )Q
(c)
w = Jc(w
′t′wt)
by the second assertion of Lemma 3.3. Hence Jc is a homomorphism of algebras. Now
Jc(Jc−1(wt)) = Jc
(
wtQ(c−1)
w
)
= wtQ(c−1)
w Q(c)
w = wt
by the first assertion of Lemma 3.3. This proves part (a) of the lemma.
Mystic Reflection Groups 7
Prove (b). In view of Lemma 3.2, it suffices to show that Q
(c)
w .+ x
k = φ
(c)
w (k)xk. Indeed,
Q(c)
w .+ x
k =
1
4
((
c+ c−1
)(
1− (−1)ki+kj
)
+
(
c− c−1 + 2
)
(−1)ki +
(
c−1 − c+ 2
)
(−1)kj
)
xk
= φ
(c)
ij (k)xk.
Finally,
Jc(wt) .+ x
k =
(
wtQ(c)
w
)
.+ x
k = (wt) .+
(
Q(c)
w .+ x
k
)
= φ(c)w (k)(wt) .+ x
k = wt .c x
k. �
Taking c = 1 in Lemma 3.4(b), we settle Proposition 2.3(d). Proposition 2.3 is proved.
Proof of Theorem 2.5
We retain the notation from the proof of Proposition 2.3. Let G = G(m, p, n) as in the theorem,
and denote T := G ∩ (C×)n. Let eT =
∑
t∈T
t ∈ CT ⊂ CG. Clearly, t
(−1)
i eT = t
(−1)
1 eT for all
i = 1, . . . , n. Hence Q
(c)
ij eT = t
(−1)
1 eT for all c ∈ C×, so that Q
(c)
w eT = t
(detw)
1 eT for all w ∈ Sn.
Since, as sets, G = {wt |w ∈ Sn, t ∈ T}, one has eG =
∑
w∈Sn
w · eT and
Jc(eG) =
∑
w∈Sn
Jc(w)eT =
∑
w∈Sn
wQ(c)
w eT =
∑
w∈Sn
wt
(detw)
1 eT = eµ(G),
because, as sets, µ(G) =
{
wt
(detw)
1 t |w ∈ Sn, t ∈ T
}
. Since a subgroup G′ of Gn is uniquely
determined by eG′ ∈ CGn, the group µ(G) is a unique subgroup G′ of Gn such that Jc(eG) = eG′ .
Finally, by Lemma 3.4(b), ρc(eG) = ρ+(eµ(G)). Setting c = 1 and using injectivity of ρ+, see
Remark 2.4, completes the proof of Theorem 2.5.
Proof of Theorem 2.8
Let G = G(m, p, n). Consider the restriction of Ji, i =
√
−1 to RG = Sn · RT where T =
G ∩ (C×)n = µ(G) ∩ (C×)n. Observe that Ji(T ) = T and
Ji(si) = siQ
(i)
si = siQ
(i)
i,i+1 =
1
2
si
(
(1 + i)t
(−1)
i + (1− i)t
(−1)
i+1
)
=
1 + i
2
σi +
1− i
2
σ−1i ,
where σi = sit
(−1)
i ∈ µ(G). Thus, Ji(RG) ⊆ Rµ(G), so that the automorphism Ji of CGn
restricts to an isomorphism RG
∼−→ Rµ(G). Theorem 2.8 is proved.
Proof of Theorem 2.11
It is convenient to prove Theorem 2.11 before Theorems 2.9 and 2.12. We start with the following
lemma.
Lemma 3.5. Let G be a thick subgroup of G(m, 1, n) = Cn o Sn where C is the subgroup of C×
of order m. Then the group T = G ∩ Cn is of the form TC,C′ = {t ∈ Cn : det t ∈ C′} for some
subgroup C′ ⊂ C of C×, and is generated by
{
t
(ε′)
1 : ε′ ∈ C′
}
∪
{
t
(ε)
i t
(ε−1)
j : ε ∈ C, 1 ≤ i, j ≤ n
}
.
Proof. Let ε be a generator of C, so that t
(ε)
1 ∈ G(m, 1, n). Since G is thick, there is an element
in G of the form ts1 where t ∈ Cn. By the normality of G, t
(ε)
1 (ts1)t
(ε−1)
1 (ts1)
−1 = t
(ε)
1 t
(ε−1)
2
belongs to G, hence to T . Because Sn acts on T (by conjugation within G), it follows that
t
(ε)
i t
(ε−1)
j ∈ T . These elements generate the subgroup T0 = TC,{1} of T .
8 Y. Bazlov and A. Berenstein
Every element t′ ∈ T is equal, modulo T0, to an element of the form t
(ε′)
1 for some ε′ ∈ C,
where ε′ = det t′. Denote by C′ the group formed by all such ε′. Then T ⊆ TC,C′ , and, since
{t(ε
′)
1 : ε′ ∈ C′} ∪ T0 generates TC,C′ , one has T ⊇ TC,C′ . �
We continue the proof of Theorem 2.11. Let G be a thick subgroup of G(m, 1, n) = Cn o Sn
so that G ∩ Cn = TC,C′ as in Lemma 3.5. Because G is thick, G contains an element of the
form ts1 where t ∈ Cn. Premultiplying ts1 by an element of TC,{1}, we conclude that G 3 t(ε)1 s1,
hence G contains (t
(ε)
1 s1)
2 = t
(ε)
1 t
(ε)
2 . It follows that t
(ε)
1 t
(ε)
2 ∈ TC,C′ , hence det t
(ε)
1 t
(ε)
2 = ε2 ∈ C′.
This means that
• either ε ∈ C′, implying t
(ε)
1 ∈ G and s1 ∈ G;
• or −ε ∈ C′, implying t
(−ε)
1 ∈ G and t
(−1)
1 s1 ∈ G.
We note the following easy lemma.
Lemma 3.6. Let G be a normal subgroup of Cn o Sn = G(m, 1, n) such that G ∩ Cn = TC,C′.
(a) If G 3 s1, then G contains all elements of the form tw with t ∈ TC,C′ and w ∈ Sn.
(b) If G 3 t(−1)1 s1, then G contains all elements tt
(detw)
1 w with t ∈ TC,C′ and w ∈ Sn.
To continue the proof of Theorem 2.11, suppose s1 ∈ G. Then by Lemma 3.6, G ⊇ Sn,
Hence G = (G∩ Cn)oSn = TC,C′ oSn. Then G = G(m, p, n) is a complex reflection group with
m
p = |C′|.
The only remaining case is s1 /∈ G but t
(−1)
1 s1 ∈ G. It follows from Lemma 3.6 that t′w ∈ G
(where t′ ∈ Cn and w ∈ Sn), if and only if t′t
(detw)
1 ∈ G, if and only if t′t
(detw)
1 ∈ G ∩ Cn = TC,C′ ,
if and only if det
(
t′t
(detw)
1
)
∈ C′. Observing that det
(
t′t
(detw)
1
)
= det(t′w), we obtain
G = {t′w ∈ Cn o Sn : det(t′w) ∈ C′} = WC,C′ .
In this case, t
(−1)
1 s1 ∈ Cn o Sn means that −1 ∈ C. That is, m is even. Theorem 2.11 is proved.
Proof of Theorem 2.9
We will use the following notion.
Definition 3.7. A thick subgroup G of G(m, 1, n) is regular, if for each normal abelian sub-
group N of G, either N = TG := G ∩ (C×)n or |N | < |TG|. Otherwise, G is singular.
Lemma 3.8.
(a) Suppose that Gi is a regular thick subgroup of G(mi, 1, ni), i = 1, 2. If G1 and G2 are
isomorphic (as abstract groups), then m1 = m2, n1 = n2 and TG1 = TG2.
(b) A thick subgroup G of G(m, 1, n) is singular, if and only if G belongs to the following list:
G(1, 1, n) with n = 2, 3, 4, G(2, 1, 2), G(2, 2, 2), µ(G(2, 2, 2)).
Proof. (a) Since Gi is regular and TGi is the unique largest order normal abelian subgroup of Gi,
the restriction of any isomorphism f : G1 → G2 to TG1 is an isomorphism TG1
∼−→ TG2 , and f
induces an isomorphism f : Sn1 = G1/TG1 → Sn2 = G2/TG2 . Hence n1 = n2. Furthermore, mi
is the exponent of the group TGi , hence m1 = m2. Finally, TGi = TC,C′i by Lemma 3.5, and
|TGi | = mni−1
i |C′i| implies C′1 = C′2. This proves part (a).
(b) Clearly, the subgroup T := TG = TC,C′ as in Lemma 3.5 is a normal abelian subgroup
of G. Let N be a normal abelian subgroup of G. We will show that N ⊆ T .
Mystic Reflection Groups 9
The map π : G→ Sn is surjective as G is thick, therefore π(N) is a normal abelian subgroup
of Sn. Hence if n = 1 or n ≥ 5, π(N) can only be {1} so N ⊆ kerπ = T and G is regular.
Let 2 ≤ n ≤ 4. Then Sn has a unique normal abelian subgroup K which is not {1}. Assume
that N 6⊂ T . Then π(N) = K. One can check that K acts on the indices 1, . . . , n transitively,
hence the centralizer of K in T is the set of scalar matrices in T , which is the center C(G) of G.
Because the conjugation action of N on T factors through π(N), we have CG(N) ∩ T = C(G).
But N is abelian, so N ⊆ CG(N). Thus, N ∩ T ⊂ C(G).
Now let wt ∈ N where 1 6= w ∈ Sn and t ∈ (C×)n. Let i ∈ {1, . . . , n} be such that w(i) 6= i,
and let ε be a generator of C. Then N 3 t(ε)i · wt · t
(ε−1)
i · (wt)−1 = t
(ε)
i t
(ε−1)
w(i) which is a scalar
matrix only if m = 1 or m = n = 2. �
To prove the if part of Theorem 2.9, let m,n be even, m
p be odd, and C′, C be subgroups
of C× of order m
p ,m, respectively. Assume for contradiction that there is an isomorphism
ϕ : WC,C′ → G(m, p, n).
In the case G = G(2, 2, 2) = T{±1},{1} × S2, G is a Klein 4-group which is not isomorphic to
W{±1},{1}, a cyclic group generated by s1t
(−1)
1 of order 4.
In all other cases, by Lemma 3.5 T = TC,C′ is the unique maximal normal abelian subgroup
of WC,C′ and of G = G(m, p, n), hence ϕ(T ) = T and ϕ induces an isomorphism ϕ : Sn =
µ(G)/T → Sn = G/T . Let us state three easy lemmas, in which we refer to a cycle of length n
in Sn as a long cycle.
Lemma 3.9. Let n be even and c ∈ Sn be a long cycle. Then det
(
t
(−1)
1 c
)
= 1 and t
(−1)
1 c ∈WC,C′.
Lemma 3.10. For t ∈ (C×)n and a long cycle c, (tc)n = z(det t), where z(ε) denotes t
(ε)
1 t
(ε)
2 · · · t
(ε)
n .
Lemma 3.11. The image of a long cycle in Sn under any automorphism of Sn is a long cycle.
Let c be a long cycle in Sn. By Lemma 3.9, c = t
(−1)
1 c ∈ µ(G). By Lemma 3.10, (c)n = z(−1).
Hence (c)n|C
′| =
(
z(−1)
)|C′|
= z(−1) as |C′| = m
p is odd.
Now consider ϕ(c) ∈ G. It is of the form tϕ(c), where t is some element of Cn such that, by
definition of G = G(m, p, n), det t ∈ C′. By Lemma 3.11, ϕ(c) is a long cycle in Sn. Therefore,
by Lemma 3.10,
(ϕ(c))n|C
′| =
(
z(det t)
)|C′|
= z((det t)
|C′|) = 1.
This is a contradiction, because the image of z(−1) 6= 1 under an isomorphism ϕ cannot be 1.
The if part of Theorem 2.9 is proved.
To establish the only if part of Theorem 2.9, observe that if m
p is even, then the groups
G = G(m, p, n) and µ(G) = WC,C′ simply coincide as subgroups of Gn. Indeed, in this case
detw ∈ {±1} ⊆ C′, hence the conditions det t ∈ C′ and det tw ∈ C′ are equivalent.
If m is even, m
p is odd and n is odd, both G = G(m, p, n) and µ(G) are normal subgroups
of G̃ = G(m, p2 , n) which do not contain the central subgroup Z =
{
1, z(−1)
}
of G̃. Hence
G̃ = G× Z = µ(G)× Z, so that G ∼= µ(G) ∼= G̃/Z. Theorem 2.9 is proved.
Proof of Theorem 2.12
Let G ⊆ G(m, 1, n), G′ ⊆ G(m′, 1, n′) be thick subgroups.
Assume that G 6= G′ are regular in the sense of Definition 3.7. Then by Lemma 3.8(a),
G ∼= G′ implies n = n′, m = m′ and TG = TG′ , so we are done by Theorem 2.11.
By inspection of the list in Lemma 3.8(b), no two singular subgroups are isomorphic to each
other. Also by inspection, one shows that if n = n′, G is singular and G′ is regular, then G
and G′ are never isomorphic.
10 Y. Bazlov and A. Berenstein
The only remaining case is when n 6= n′, G is singular and G′ is regular. If G ∼= G′, then
n, n′ ≤ 4, because |S5| > |G|. Then, an easy analysis based on the cardinalities of singular
groups shows that the only possible isomorphism is the one given in part (b). Theorem 2.12 is
proved.
4 Appendix
The aim of this section is to prove the following important result about group ring actions.
Theorem 4.1. Let A be an integral domain and G be a group of ring automorphisms of A.
Then:
(a) the natural map ρ : AG→ EndZ(A) given by
(
ρ
(∑
i
aigi
))
(a) =
∑
i
aigi(a) is injective;
(b) with respect to the natural ring structure on EndZ(A) and the semidirect product structure
Ao ZG on AG, the map ρ is a ring homomorphism.
The following is immediate from the theorem.
Corollary 4.2. In the notation of the theorem, let R be a subring of AG. Then the restriction
of ρ to RG is an injective ring homomorphism
RG ↪→ EndRA,
where RG is the ordinary group ring of G.
Proof of Theorem 4.1. We need the following generalization of the celebrated Dedekind’s
lemma.
Lemma 4.3. Let A be an integral domain and B be a multiplicative monoid. Let G be a set of
monoid homomorphisms B → A. Then the natural A-linear map
ρ : AG → Fun(B,A),
(
ρ
(∑
g
agg
))
(b) =
∑
g
agg(b)
is injective. (Here AG = ⊕g∈GAg is the free A-module generated by G.)
Proof. Assume for contradiction that ρ is not injective. Let G0 be a minimal finite subset of G
such that there exists a non-zero element
∑
g∈G0
agg in the kernel of ρ. That is,
∑
g∈G0
agg(b) = 0 for all b ∈ B.
Clearly, all ag are non-zero and |G0| > 1 because A has no zero divisors. In particular, for each
b, b′ ∈ B,∑
g∈G0
agg(b′b) =
∑
g∈G0
agg(b′)g(b) = 0,
because all g ∈ G0 are homomorphisms from B to A. Furthermore, fix h ∈ G0. Combining the
above identities, we obtain∑
g∈G0\{h}
agg(b′)(g(b)− h(b)) = 0 for all b, b′ ∈ B.
Mystic Reflection Groups 11
That is, for each b ∈ B, the element
kb :=
∑
g∈G0\{h}
ag(g(b)− h(b))g
belongs to ker ρ. The minimality of G0 implies that kb = 0 for all b ∈ B, which is equivalent to
g(b) = h(b) for all b ∈ B and all g ∈ G0. That is, |G0| = 1, a contradiction. �
We now use Lemma 4.3 for A as in Theorem 4.1 with B = A \ {0} and G = G viewed as
homomorphisms of monoids A \ {0} → A. Since EndZA is naturally a subset of Fun(A \ {0}, A)
and ρ(AG) ∈ EndZA ⊂ Fun(A \ {0}, A), part (a) of Theorem 4.1 is proved.
To prove (b), note that EndZA is naturally a ring, with multiplication being composition
of maps. The semidirect product multiplication on AG is given by the formula (ag)(a′g′) =
ag(a′) · gg′ for all a, a′ ∈ A, g, g′ ∈ G. Then
ρ((ag)(a′g′))(b) = a · g(a′ · g′(b)) = a · g(a′) · g(g′(b)) = ag(a′)(gg′)(b)
for all a, a′, b ∈ A, g, g′ ∈ G. Part (b) of the theorem is proved. �
Acknowledgments
We thank Ken Brown for bringing the paper [2] to our attention, and Alexander Premet for
stimulating discussions. The present paper was started when both authors were research mem-
bers of the Mathematical Sciences Research Institute. We thank the Institute and the organizers
of the Noncommutative Algebraic Geometry and Representation Theory program for creating
an atmosphere conducive for research. We acknowledge partial support of the LMS Research
in Pairs grant ref. 41224. The second named author was partially supported by the NSF grant
DMS-1101507.
References
[1] Bazlov Y., Berenstein A., Noncommutative Dunkl operators and braided Cherednik algebras, Selecta
Math. (N.S.) 14 (2009), 325–372, arXiv:0806.0867.
[2] Kirkman E., Kuzmanovich J., Zhang J.J., Shephard–Todd–Chevalley theorem for skew polynomial rings,
Algebr. Represent. Theory 13 (2010), 127–158, arXiv:0806.3210.
http://dx.doi.org/10.1007/s00029-009-0525-x
http://dx.doi.org/10.1007/s00029-009-0525-x
http://arxiv.org/abs/0806.0867
http://dx.doi.org/10.1007/s10468-008-9109-2
http://arxiv.org/abs/0806.3210
1 Introduction
2 Main results
3 Proofs of results from Section 2
4 Appendix
References
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