Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy
We prove that multiparameter Schur Q-functions, which include as specializations factorial Schur Q-functions and classical Schur Q-functions, provide solutions of the BKP hierarchy.
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| Цитувати: | Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy / N. Rozhkovskaya // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 19 назв. — англ. |
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| author_facet | Rozhkovskaya, N. |
| citation_txt | Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy / N. Rozhkovskaya // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 19 назв. — англ. |
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| description | We prove that multiparameter Schur Q-functions, which include as specializations factorial Schur Q-functions and classical Schur Q-functions, provide solutions of the BKP hierarchy.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 065, 14 pages
Multiparameter Schur Q-Functions
Are Solutions of the BKP Hierarchy
Natasha ROZHKOVSKAYA
Department of Mathematics, Kansas State University, Manhattan, KS 66502, USA
E-mail: rozhkovs@math.ksu.edu
URL: http://www.math.ksu.edu/~rozhkovs/
Received May 20, 2019, in final form August 23, 2019; Published online August 28, 2019
https://doi.org/10.3842/SIGMA.2019.065
Abstract. We prove that multiparameter Schur Q-functions, which include as specializa-
tions factorial Schur Q-functions and classical Schur Q-functions, provide solutions of the
BKP hierarchy.
Key words: BKP hierarchy; symmetric functions; factorial Schur Q-functions; multiparame-
ter Schur Q-functions; vertex operators
2010 Mathematics Subject Classification: 05E05; 17B65; 17B69; 11C20
1 Introduction
Integrable systems of the KP (Kadomtsev–Petviashvilli) type hierarchy of partial differential
equations, which corresponds to infinite-dimensional Lie algebra of type A, and of its type B
variant, the BKP hierarchy, have as solutions renowned families of symmetric functions – Schur
polynomials in the KP case, and Schur Q-polynomials in the BKP case [2, 3, 4, 10, 14, 17, 18, 19],
etc. In this note we show that multiparameter Schur Q-functions also provide solutions of the
BKP hierarchy.
Multiparameter Schur Q-functions Q
(a)
λ were introduced and studied combinatorially in [8].
These symmetric functions are interpolation analogues of the classical Schur Q-functions de-
pending on a sequence of complex valued parameters a = (a0, a1, . . . ). The definition of mul-
tiparameter Schur Q-functions is reproduced in (7.1). Classical Schur Q-functions correspond
to a = (0, 0, 0, . . . ), and with the evaluation a = (0, 1, 2, 3, . . . ) the multiparameter Schur Q-
functions are called factorial Schur Q-functions. These families of symmetric functions proved
to be useful in study of a number of questions of representation theory and algebraic geometry.
Here are a few examples.
The authors of [1, 15, 16] described Capelli polynomials of the queer Lie superalgebra which
form a distinguished family of super-polynomial differential operators indexed by strict partitions
acting on an associative superalgebra. The eigenvalues of these Capelli polynomials are expressed
through the factorial Schur Q-functions.
In [5, 7] the equivariant cohomology of a Lagrangian Grassmannian of a symplectic or or-
thogonal types is studied. The restrictions of Schubert classes to the set of points fixed under
the action of a maximal torus of the symplectic group are calculated in terms of factorial sym-
metric functions. Further in [6] factorial Schur Q-functions are used to write generators and
relations for the equivariant quantum cohomology rings of the maximal isotropic Grassmannians
of types B, C and D.
This paper is a contribution to the Special Issue on Representation Theory and Integrable Systems in honor
of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday. The full collection is
available at https://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html
mailto:rozhkovs@math.ksu.edu
http://www.math.ksu.edu/~rozhkovs/
https://doi.org/10.3842/SIGMA.2019.065
https://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html
2 N. Rozhkovskaya
In [12] the center of the twisted version of Khovanov’s Heisenberg category is identified with
the algebra generated by classical Schur Q-functions (denoted as Bodd in the exposition below).
Factorial Schur Q-functions are described as closed diagrams of this category.
The goal of this note is to show that multiparameter Schur Q-functions Q
(a)
λ are solutions of
the BKP hierarchy. The origin for this phenomena lies in the fact proved in [11] that generating
functions of multiparameter Schur Q-functions and of classical Schur Q-functions coincide.
While the BKP hierarchy is described in a wide range of literature on integrable systems and
solitons, for the completeness of exposition and for the convenience of the reader we formulate
the whole setting of the BKP hierarchy in terms of generating functions of symmetric functions
with the neutral fermions bilinear identity (5.1) as a starting point. We avoid to use any other
facts than the well-known properties of symmetric functions that can be found in the classical
monograph [13], and through the text we provide the references to the corresponding chapters
and examples of that monograph.
It is worth to mention that formulation of the KP and the BKP integrable systems solely
in terms of symmetric functions can be found, e.g., in [9]. The authors of [9] start with the
bilinear identities in integral form, then, using the Cauchy type orthogonality properties of
symmetric functions (cf. [13, Chapter III, equation (8.13)]), they arrive at Plucker type relations,
and the later ones are transformed into the collection of partial differential equations of Hirota
derivatives that constitute the hierarchy. As it is mentioned above, our route is traced differently
employing the properties of generating functions of complete, elementary symmetric functions
and power sums. We obtain differential equations of the hierarchy in Hirota form as coefficients
of Taylor expansions. One of the advantages of this approach is that it directly addresses
the corresponding vertex operators actions, since the later ones are also ‘generating functions’
(formal distributions).
The paper is organized as follows. In Section 2 we recall some facts about complete, elemen-
tary symmetric functions, power sums and classical Schur Q-functions. In Section 3 we describe
the action of neutral fermions on the space generated by classical Schur Q-functions. In Section 4
we review properties of generating functions for multiplication operators and corresponding ad-
joint operators and deduce vertex operator form of the formal distribution of neutral fermions.
In Section 5 we review all the steps of recovering the BKP hierarchy of partial differential equa-
tions in Hirota form from the neutral fermions bilinear identity. In Section 6 we make simple
observation that immediately shows that classical Schur Q-functions are solutions of the BKP
hierarchy (which recovers the result of [18]). In Section 7 we introduce multiparameter Schur
Q-functions, and using the observation of Section 6, we show that Q
(a)
λ are also solutions of the
BKP hierarchy.
2 Schur Q-functions
Let B be the ring of symmetric functions in variables (x1, x2, . . . ). Consider the families of the
following symmetric functions:
elementary symmetric functions
ek =
∑
i1<···<ik
xi1 · · ·xik | k = 0, 1, . . .
,
complete symmetric functions
hk =
∑
i1≤···≤ik
xi1 · · ·xik | k = 0, 1, . . .
,
symmetric power sums
{
pk =
∑
xki | k = 0, 1, . . .
}
.
Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy 3
We set ek = hk = 0 for k < 0. It is well-known [13, Chapter I.2], that each of these families
generate B as a polynomial ring:
B = C[p1, p2, p3, . . . ] = C[e1, e2, e3, . . . ] = C[h1, h2, h3, . . . ].
Combine the families hk, ek, pk into generating functions
H(u) =
∑
k≥0
hk
uk
, E(u) =
∑
k≥0
ek
uk
, P (u) =
∑
k≥1
pk
uk
.
The following facts are well-known [13, Chapter I.2].
Lemma 2.1.
H(u) =
∏
i
1
1− xi/u
, E(u) =
∏
i
1 + xi/u, E(−u)H(u) = 1,
H(u) = exp
∑
n≥1
1
n
pn
un
, E(u) = exp
∑
n≥1
(−1)n−1
n
pn
un
.
We introduce one more family of symmetric functions {Qk = Qk(x1, x2, . . . )} with (k =
0, 1, . . . ) as the coefficients of the generating function
Q(u) =
∑
k≥0
Qk
uk
=
∏
i
u+ xi
u− xi
. (2.1)
From Lemma 2.1 and (2.1) we immediately get relations of the next lemma.
Lemma 2.2.
Q(u) = E(u)H(u) = R(u)2, R(u) = exp
∑
n∈Nodd
pn
nun
,
where Nodd = {1, 3, 5, . . . }.
Note that Q(u)Q(−u) = 1, which implies that Qr with even r can be expressed algebraically
through Qr with odd r:
Q2m =
m−1∑
r=1
(−1)r−1QrQ2m−r +
1
2
(−1)m−1Q2
m.
More generally, Schur Q-functions Qλ labeled by strict partitions are defined as a specialization
of Hall–Littlewood polynomials [13, Chapter III.2].
Definition 2.3. Let λ = (λ1 > λ2 > · · · > λl) be a strict partition. Let l ≤ N . Schur Q-
polynomial Qλ(x1, . . . , xN ) is the symmetric polynomial in variables xi’s defined by the formula
Qλ(x1, . . . , xN ) =
2l
(N − l)!
∑
σ∈SN
l∏
i=1
xλiσ(i)
∏
i<j
xσ(i) + xσ(j)
xσ(i) − xσ(j)
. (2.2)
4 N. Rozhkovskaya
Alternatively, Schur Q-polynomial Qλ = Qλ(x1, . . . , xN ) for N > l is the coefficient of
u−λ1 · · ·u−λl in the formal generating function
Q(u1, . . . , ul) =
∑
λ1,...,λl∈Z
Qλ
uλ1 · · ·uλl
=
∏
1≤i<j≤l
uj − ui
uj + ui
l∏
i=1
Q(ui), (2.3)
where it is understood that
uj − ui
uj + ui
= 1 + 2
∑
r≥1
(−1)ruriu
−r
j ,
and Q(u) is given by (2.1) [13, Chapter III, equation (8.8)]. Schur Q-polynomials have a sta-
bilization property, hence, one can omit the number N of variables x′is as long as it is not less
than the length of the partition λ and consider Qλ as functions of infinitely many variables
(x1, x2, . . . ).
3 Action of neutral fermions on bosonic space Bodd
Consider the subalgebra Bodd of B generated by odd ordinary Schur Q-functions: Bodd =
C[Q1, Q3, . . . ]. It is known that Bodd is also a polynomial algebra in odd power sums Bodd =
C[p1, p3, . . . ] and that Schur Q-functions Qλ labeled by strict partitions constitute a linear basis
of Bodd [13, Chapter III.8, equation (8.9)].
Define operators {ϕk}k∈Z acting on the coefficients of generating functions Q(u1, . . . , ul) by
the rule
Φ(v)Q(u1, . . . , ul) = Q(v, u1, . . . , ul) (3.1)
with Φ(v) =
∑
m∈Z
ϕmv
−m. Then in the expansion (2.3)
ϕm : Qλ 7→ Q(m,λ).
Observe that from (2.3)
(Φ(u)Φ(v) + Φ(v)Φ(u))Q(u1, . . . , ul)
= 2
1 +
∑
r≥1
(−1)rurv−r +
∑
r≥1
(−1)rvru−r
A(u, v, u1, . . . , ul)
= 2
∑
r∈Z
(
−u
v
)r
A(u, v, u1, . . . , ul),
where
A(u, v, u1, . . . , ul) =
∏
1≤j≤l
(uj − u)(uj − v)
(uj + u)(uj + v)
Q(u)Q(v)Q(u1, . . . , ul).
Using that δ(u, v) =
∑
r∈Z
urv−(r+1) is a formal delta distribution with the property δ(u, v)a(u) =
δ(u, v)a(v) for any formal distribution a(u) =
∑
n∈Z
anu
n, and that Q(u)Q(−u) = 1, we get
(Φ(u)Φ(v) + Φ(v)Φ(u))Q(u1, . . . , ul) = 2vδ(−u, v)Q(u1, . . . , ul). (3.2)
Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy 5
Since coefficients of the expansion of Q(u1, . . . , ul) in powers of u1, . . . , ul include Schur Q-
functions Qλ, and the latter form a linear basis of Bodd, it follows that (3.1) provides the action
of well-defined operators {ϕk}k∈Z on Bodd:
ϕk(Qλ) = Q(k,λ).
Relation (3.2) on generating functions is equivalent to the commutation relations
[ϕm, ϕn]+ = 2(−1)mδm+n,0 for m,n ∈ Z. (3.3)
Thus, operators {ϕi}i∈Z and 1 provide the action of Clifford algebra Clϕ of neutral fermions on
the Fock space Bodd. Note that for any strict partition λ = (λ1 > λ2 > · · · > λl)
Q(λ1,...λl) = ϕλ1 · · ·ϕλl(1), (3.4)
or in terms of generating functions,
Q(u1, . . . , ul) = Φ(u1) · · ·Φ(ul)(1). (3.5)
Formulae (3.4), (3.5) sometimes are called the vertex operator realization of Schur Q-functions.
4 Vertex operator form of formal distribution
of neutral fermions
It will be convenient for us to consider Bodd as a subring of the ring of symmetric functions B.
This allows us to recover the celebrated vertex operator form of the formal distribution of
neutral fermions Φ(u) from no-less celebrated properties of generating functions of complete
and elementary symmetric functions. All of these properties are discussed in [13, Chapter I].
The ring of symmetric functions B possesses a bilinear form (·, ·) [13, Chapter I, equation (4.5)]
defined on the linear basis of monomials of power sums labeled by partitions λ and µ as
(pλ1 · · · pλl , pµ1 · · · pµl) = zλδλ,µ,
where zλ =
∏
imimi! and mi = mi(λ) is the number of parts of λ equal to i.
We will use this form and its restriction to Bodd to define adjoint operators1 of the multipli-
cation operators. By definition, given an element f ∈ B, the operator f⊥ adjoint to the operator
of multiplication by f is given by the rule(
f⊥g, h
)
= (g, fh) for any g, h ∈ B.
[13, Chapter I.5, Example 3] contains the following statement. Consider a symmetric function
f = f(p1, p2, . . . ) expressed as a polynomial in power sums pi. Then the adjoint operator on B
to the multiplication operator by f is given by
f⊥ = f
(
∂
∂p1
,
2∂
∂p2
,
3∂
∂p3
, . . .
)
. (4.1)
In particular p⊥n = n∂/∂pn.
1Traditionally, one uses rescaled form on Bodd defined as (pλ, pµ) = 2−l(λ)zλδλ,µ, where l(λ) is the number of
parts of λ, but rescaling is not necessary for our purposes, since in the rescaled form p⊥n = n/2 · ∂/∂pn (see [13,
Chapter III.8, Example 11]).
6 N. Rozhkovskaya
Combine the corresponding adjoint operators of the families hk, ek, pk and Qk into generating
functions
H⊥(u) =
∑
k≥0
h⊥k u
k, E⊥(u) =
∑
k≥0
e⊥k u
k,
P⊥(u) =
∑
k≥1
p⊥k uk, Q⊥(u) =
∑
k≥0
Q⊥k uk.
Then (4.1) immediately implies the following relations.
Lemma 4.1.
H⊥(u) = exp
∑
n≥1
∂
∂pn
un
, E⊥(u) = exp
∑
n≥1
(−1)n−1
∂
∂pn
un
,
Q⊥(u) = E⊥(u)H⊥(u) = R⊥(u)2, R⊥(u) = exp
∑
n∈Nodd
∂
∂pn
un
,
where Nodd = {1, 3, 5, . . . }.
The proof of the next lemma is outlined in [13, Chapter I.5, Example 29].
Lemma 4.2. The following commutation relations on generating functions of multiplication and
adjoint operators acting on B hold:
H⊥(u) ◦H(v) = (1− u/v)−1H(u) ◦H⊥(v),
H⊥(u) ◦ E(v) = (1 + u/v)E(u) ◦H⊥(v),
E⊥(u) ◦H(v) = (1 + u/v)H(u) ◦ E⊥(v),
E⊥(u) ◦ E(v) = (1− u/v)−1E(u) ◦ E⊥(v).
Corollary 4.3.
H⊥(u) ◦Q(v) =
v + u
v − u
Q(u) ◦H⊥(v),
E⊥(u) ◦Q(v) =
v + u
v − u
Q(u) ◦ E⊥(v),
R⊥(u) ◦Q(v) =
v + u
v − u
Q(u) ◦R⊥(v).
Proof. For the first and second one we use that Q(u) = E(u)H(u). Observe that
H⊥(u)|Bodd = E⊥(u)|Bodd = R⊥(u)|Bodd .
In other words, since Qk does not depend on even power sums p2r, we can add terms ∂/∂p2r in
the sum under the exponent when applying to elements of Bodd:
R(u)⊥(Q(v)) = exp
∑
n∈Nodd
∂
∂pn
un
Q(v) = exp
∑
n≥1
∂
∂pn
un
Q(v) = H⊥(u)Q(v). �
We arrive at the vertex operator form of formal distribution of neutral fermions.
Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy 7
Proposition 4.4.
Φ(v) = Q(v)R(−v)⊥ = exp
∑
n∈Nodd
2pn
n
1
vn
exp
− ∑
n∈Nodd
∂
∂pn
vn
. (4.2)
Proof. From Corollary 4.3, the action of the operator Q(v)R(−v)⊥ on the coefficients of gen-
erating function Q(u1, . . . , ul) coincides with the action of Φ(v):
Q(v)R(−v)⊥(Q(u1, . . . , ul)) = Q(v)
∏
1≤i<j≤l
uj − ui
uj + ui
R(−v)⊥
(
l∏
i=1
Q(ui)
)
= Q(v)
∏
1≤j<i≤l
uj − ui
uj + ui
l∏
i=1
v − ui
v + ui
l∏
i=1
Q(ui) = Q(v, u1, . . . , ul).
Since coefficients of Q(u1, . . . , ul) contain a linear basis of Bodd, the equality (4.2) follows. �
5 The neutral fermions bilinear identity
Let
Ω =
∑
n
ϕn ⊗ (−1)nϕ−n.
One looks for the solutions in Bodd of the neutral fermions bilinear identity
Ω(τ ⊗ τ) = τ ⊗ τ, (5.1)
where τ = τ(p̃) = τ(2p1, 2p3/3, 2p5/5, . . . ). It is known [2, 3, 4, 10] that (5.1) is equivalent to
an infinite integrable system of partial differential equations called the BKP hierarchy. Further
in Section 6 a simple observation explains, why Schur Q-functions constitute solutions of the
neutral fermions bilinear identity, and hence of the BKP hierarchy.
In this section we would like to make a small deviation and review the steps of recovering
the BKP hierarchy of partial differential equations in the Hirota form from the neutral fermions
bilinear identity. This is certainly a well-known procedure. However, the explicit calculations
are often omitted in the literature, and we would like to provide them here for the convenience
of the reader.
Note that Ω is the constant coefficient of the formal distribution Φ(u)⊗Φ(−u), or, in terms
of residue,
Ω = Res
u=0
1
u
Φ(u)⊗ Φ(−u). (5.2)
We identify Bodd⊗Bodd with C[p1, p3, . . . ]⊗C[r1, r3, . . . ] – two copies of polynomial rings, where
variables in each of them play role of power sum symmetric functions. Set p̃ = (2p1, 2p3/3,
2p5/5, . . . ), r̃ = (2r1, 2r3/3, 2r5/5, . . . ). Then ∂pn = 2∂p̃n/n and
Φ(u)τ ⊗ Φ(−u)τ = exp
∑
n∈Nodd
(p̃n − r̃n)
1
un
× exp
− ∑
n∈Nodd
2
n
(
∂
∂p̃n
− ∂
∂r̃n
)
un
τ(p̃)τ(r̃).
8 N. Rozhkovskaya
Introduce the change of variables
p̃n = xn − yn, r̃n = xn + yn.
Then
Φ(u)τ ⊗ Φ(−u)τ = exp
∑
n∈Nodd
−2yn
1
un
exp
∑
n∈Nodd
2
n
∂
∂yn
un
τ(x− y)τ(x+ y)
with (x± y) = (x1 ± y1, x3 ± y3, x5 ± y5, . . . ).
Definition 5.1. Let P (D) be a multivariable polynomial in the collection of variables D =
(D1, D2, . . . ), let f(x), g(x) be differentiable functions in x = (x1, x2, . . . ).
The Hirota derivative P (D)f ·g is a function in variables (x1, x2, . . . ) given by the expression
P (D)f · g = P (∂z1 , ∂z1 , . . . )f(x+ z)g(x− z)|z=0,
where x± z = (x1 ± z1, x2 ± z2, . . . ).
For example,
Dn
i f · g =
n∑
k=0
(−1)k
(
n
k
)
∂kf
∂xki
∂n−kg
∂xn−ki
,
which implies in particular that odd Hirota derivatives are tautologically zero when f = g:
D2n+1
i f · f = 0 for n = 0, 1, 2, . . . .
The following lemma allows one to rewrite bilinear identity (5.1) in terms of the Hirota deriva-
tives.
Lemma 5.2.
exp
∑
n∈Nodd
2
n
∂
∂yn
un
τ(x− y)τ(x+ y) = exp
∑
n∈Nodd
(
yn +
2
n
un
)
Dn
τ · τ.
Proof. By the Taylor series expansion,
ea∂/∂yg(y) =
∞∑
n=0
ang(n)(y)
n!
= g(y + a). (5.3)
Applying (5.3) twice with t = (t1, t3, t5, . . . ), ũ =
(
2u, 2u3/3, 2u5/5, . . .
)
,
exp
∑
n∈Nodd
2
n
∂
∂yn
un
τ(x− y)τ(x+ y) = τ(x+ y + ũ)τ(x− y − ũ)
= τ(x+ y + ũ+ t)τ(x− (y + ũ+ t))|t=0
= exp
∑
n∈Nodd
(
yn +
2
n
un
)
∂
∂tn
τ(x+ t)τ(x− t)
∣∣∣∣∣∣
t=0
. �
Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy 9
Thus, we can write in terms of Hirota derivatives
Φ(u)τ ⊗ Φ(−u)τ = exp
∑
n∈Nodd
−2yn
un
× exp
∑
n∈Nodd
2
n
Dnu
n
exp
∑
n∈Nodd
ynDn
τ · τ. (5.4)
In order to compute Res
u=0
1
uΦ(u)τ ⊗ Φ(−u)τ , which is just the coefficient of u0 of Φ(u)τ ⊗
Φ(−u)τ , we recall the following well-known facts on the composition of exponential series with
generating series. Their proofs can be done, e.g., by induction, or again found in [13, Chapter I].
Proposition 5.3. Let S(u) =
∞∑
k=0
Sk
1
uk
and X(u) =
∞∑
k=1
Xk
1
uk
be related by
exp(X(u)) = S(u).
Then the following statements hold
Sk =
k∑
s=1
∑
l1+2l2+···+sls=k, li≥1
1
l1! · · · ls!
X l1
1 · · ·X
ls
s ,
Sk = det
1
n!
X1 −1 0 0 . . . 0
2X2 X1 −2 0 . . . 0
3X3 2X2 X1 −3 . . . 0
. . . . . . . . . . . . . . . 0
kXk (k − 1)Xk−1 (k − 2)Xk−2 (k − 3)Xk−3 . . . X1
,
Xk =
(−1)k−1
k
det
S1 1 0 0 . . . 0
2S2 S1 1 0 . . . 0
3S3 S2 S1 1 . . . 0
. . . . . . . . . . . . . . . 0
kSk Sk−1 Sk−2 Sk−3 . . . S1
.
Example 5.4.
S0 = 1, S1 = X1, S2 =
1
2
X2
1 +X2, S3 = S3 +X2X1 +
1
6
X3
1 ,
S4 = X4 +X3X1 +
1
2
X2
2 +
1
2
X2X
2
1 +
1
24
X4
1 .
By Lemma 2.1, when X variables in these formulae are interpreted as normalized power sums
Xk = pk/k, Sk’s are identified with complete symmetric functions hk’s.
Example 5.5. Let X2k = 0 for k = 1, 2, . . . . Then the first few Sn = Sn(X1, 0, X3, . . . ) are
given by
S0 = 1, S1 = X1, S2 =
1
2
X2
1 , S3 = X3 +
1
6
X3
1 , S4 = X3X1 +
1
24
X4
1 ,
S5 =
1
120
X5
1 +
1
2
X2
1X3 +X5, S6 =
1
720
X6
1 +
1
6
X3
1X3 +
1
2
X2
3 +X1X5,
S7 =
1
5040
X7
1 +
1
24
X4
1X3 +
1
2
X1X
2
3 +
1
2
X1X5 +X7.
Note that by Lemma 2.2 when X variables in these formulae are interpreted as odd normalized
power sums X2k+1 = 2p2k+1/(2k + 1), Sk’s are identified with Schur Q-functions Qk’s.
10 N. Rozhkovskaya
Using the statement of Proposition 5.3, we can write the coefficient of u0 of (5.4) as
∞∑
m=0
Sm (ỹ)Sm
(
D̃
)
exp
∑
n∈Nodd
ynDn
τ · τ = τ(x− y) · τ(x+ y), (5.5)
where ỹ = (−2y1, 0,−2y3, . . . ), D̃ = (2D1, 0, 2D3/3, 0, . . . ).
Note that S0 = 1 and exp
( ∑
n∈Nodd
ynDn
)
τ · τ = τ(x−y) · τ(x+y), hence we can rewrite (5.5)
as
∞∑
m=1
Sm(ỹ)Sm
(
D̃
)
exp
∑
n∈Nodd
ynDn
τ · τ = 0. (5.6)
To obtain the equations of the BKP hierarchy, one expands the left hand side of (5.6) in mono-
mials ym1
1 ym2
2 · · · y
mN
N to obtain as coefficients Hirota operators in terms of Dk’s.
For example, let us compute the coefficient of y23. In the expansion of(
S1(ỹ)S1
(
D̃
)
+ S2(ỹ)S2
(
D̃
)
+ S3(ỹ)S3
(
D̃
)
+ · · ·
)(
1 +
∑
yiDi +
1
2
(∑
yiDi
)2
+ · · ·
)
the term y23 appears in S3(ỹ)S3
(
D̃
)
× y3D3 and in S6(ỹ)S6
(
D̃
)
× 1. Using the expansions of
Example 5.5, the coefficient of y23 is
−2S3
(
D̃
)
D3 + 2S6
(
D̃
)
=
8
45
(
D6
1 − 5D1D3 − 5D2
3 + 9D1D5
)
,
which provides the Hirota bilinear form of the BKP equation that gives the name to the hierarchy(
D6
1 − 5D1D3 − 5D2
3 + 9D1D5
)
τ · τ = 0.
Remark 5.6. Writing the residue (5.2) as a contour integral, one gets the BKP in its integral
form ∮
1
2πiu
exp
∑
n∈Nodd
(p̃n − r̃n)
1
un
exp
− ∑
n∈Nodd
2
n
(
∂
∂p̃n
− ∂
∂r̃n
)
un
τ(p̃)τ(r̃)
= τ(p̃)τ(r̃).
6 Commutation relation for the bilinear identity
Our goal is to show that multiparameter Schur Q-functions are solutions of the neutral fermions
bilinear identity (5.1), thus they provide solutions of the BKP hierarchy.
Let X =
∑
n>0
Anϕn for some An ∈ C. From (3.3) one gets X2 = 0.
Proposition 6.1.
Ω(X ⊗X) = (X ⊗X)Ω.
Proof.
Ω(X ⊗X) =
∑
k∈Z
ϕkX ⊗ (−1)kϕ−kX
=
∑
k∈Z
(−Xϕk + [ϕk, X]+)⊗ (−1)k(−Xϕ−k + [ϕ−k, X]+).
Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy 11
Note that [ϕk, X]+ =
[
ϕk,
∑
n>0
Anϕn
]
+
= 2
∑
n>0
(−1)nAnδk+n,0, hence
Ω(X ⊗X) = (X ⊗X)Ω−
∑
n>0
2(−1)nAn ⊗ (−1)nXϕn −
∑
n>0
(−1)nXϕn ⊗ 2(−1)nAn
+ 4
∑
k∈Z
∑
m,n>0
(−1)nAnδn+k,0 ⊗ (−1)mAmδm−k,0
= (X ⊗X)Ω− 2⊗X2 −X2 ⊗ 2 + 4
∑
m,n>0
(−1)nAn ⊗ (−1)mAmδm+n,0.
We use that X2 = 0, and since both m and n in the last sum are always positive, the last term
is also zero. �
Corollary 6.2. Let τ ∈ Bodd be a solution of (5.1), and let X =
∑
n>0
Anϕn with An ∈ C. Then
τ ′ = Xτ is also a solution of (5.1).
The vertex operator presentation (3.4) of Schur Q-functions and Corollary 6.2 immediately
imply that Schur Q-functions are solutions of (5.1), since constant function 1 is a solution
of (5.1). This argument reproves the result of [18] and easily extends to more general case of
multiparameter Schur Q-functions defined in the next section.
7 Multiparameter Schur Q-functions are solutions
of the BKP hierarchy
Multiparameter Schur Q-functions were introduced in [8] as a generalization of definition (2.2).
Fix an infinite sequence of complex numbers a = (a0, a1, a2, . . . ). Consider the analogue of
a power of a variable x
(x|a)k = (x− a0)(x− a2) · · · (x− ak−1).
We also define a shift operation τ : ak 7→ ak+1, so that
(x|τa)k = (x− a1)(x− a2) · · · (x− ak).
Definition 7.1. Let α = (α1, . . . , αl) be a vector with non-negative integer coefficients αi ∈ Z≥0.
The multiparameter Schur Q-function in variables (x1, . . . , xN ) with l ≤ N is defined by
Q(a)
α (x1, . . . , xN ) =
2l
(N − l)!
∑
σ∈SN
l∏
i=1
(xσ(i)|a)αi
∏
i≤l,i<j≤N
xσ(i) + xσ(j)
xσ(i) − xσ(j)
. (7.1)
When a = (0, 0, 0, . . . ) and α is a strict partition, one gets back (2.2), the classical Schur
Q-functions Qα(x1, . . . , xN ). The evaluation a = (0, 1, 2, . . . ) gives factorial Schur Q-functions
denoted as Q∗α(x), those applications are outlined in the introduction. The multiparameter Schur
Q-functions enjoy a stability property, hence one can consider Q
(a)
α (x1, x2, . . . ) to be a function
of infinitely many variables.
Note from (7.1) that for any permutation σ ∈ Sl,
Q(a)
α (x1, . . . , xN ) = (−1)σQ
(a)
σ(α)(x1, . . . , xN ), (7.2)
where (−1)σ is the sign of permutation σ [11, Proposition 3]. Hence, Q
(a)
α = 0 if αi = αj for
some i, j, and for a vector α = (α1, . . . , αl) with positive distinct integer coefficients αi ∈ Z>0,
function Q
(a)
α coincides up to a sign with another Q
(a)
α′ labeled by strict partition α′.
One can check directly the following transitions between regular and multiparameter powers
of variables.
12 N. Rozhkovskaya
Lemma 7.2. For n = 0, 1, 2, . . .
(u− a1) · · · (u− an) =
∞∑
k=0
(−1)n−ken−k(a1, . . . , an)uk,
1
(u− a1) · · · (u− an)
=
∞∑
k=0
hk−n(a1, . . . , an)u−k,
un =
∞∑
k=0
hn−k(a1, . . . , ak+1)(u− a1) · · · (u− ak),
1
un
=
∞∑
k=0
(−1)n−kek−n(a1, . . . , ak−1)
1
(u− a1) · · · (u− ak)
.
Double application of Lemma 7.2 implies the following useful observation.
Lemma 7.3. For any sequence a = (0, a1, a2, . . . )
∞∑
m=0
(x|a)m
(u|τa)m
=
∞∑
m=0
xm
um
.
Proof.
∞∑
m=0
(x|a)m
(u|τa)m
=
∞∑
m,k=0
(−1)m−kem−k(0, a1 · · · am−1)xk
1
(u|τa)m
=
∞∑
k=0
xk
∞∑
m=0
(−1)k−mem−k(a1 · · · am−1)
1
(u|τa)m
=
∞∑
k=0
xk
uk
. �
Consider a part of the generating function (2.3) of ordinary Schur Q-functions that corre-
sponds only to positive values of λi:
Q+(u1, . . . , ul) =
∑
λ1,...,λl∈Z>0
Qλ
uλ11 · · ·u
λl
l
.
By (7.2), every non-zero coefficient of Q+(u1, . . . , ul) up to a sign coincides with a classical
Schur Q-function labeled by an appropriate strict partition. In [11] the following remarkable
observation is made.
Theorem 7.4 ([11]). For any sequence a = (0, a1, a2, . . . )
Q+(u1, . . . , ul) =
∑
λ1,...,λl∈Z>0
Q
(a)
λ
(u1|τa)λ1 · · · (ul|τa)λl
.
Proof. In [11] theorem is proved by induction on the length of the vector λ. A very short proof
of this theorem follows from Lemma 7.3 and definition (7.1). Indeed,
∑
λi∈Z>0
Q
(a)
λ
(u1|τa)λ1 · · · (ul|τa)λl
=
2l
(N − l)!
∑
σ∈SN
l∏
i=1
∑
λi∈Z>0
(xσ(i)|a)λi
(ui|τa)λi
∏
i≤l,i<j≤N
xσ(i) + xσ(j)
xσ(i) − xσ(j)
=
2l
(N − l)!
∑
σ∈SN
l∏
i=1
∑
λi∈Z>0
xλiσ(i)
uλii
∏
i≤l,i<j≤N
xσ(i) + xσ(j)
xσ(i) − xσ(j)
= Q+(u1, . . . , ul). �
Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy 13
Thus, Theorem 7.4 suggests that multiparameter Schur Q-functions are obtained from clas-
sical Schur Q-functions by the change of the basis of expansion
{
1/uk
}
→
{
1/(u|τa)k
}
in the
generating function Q+(u1, . . . , ul).
Lemma 7.2 immediately implies the following relations between classical and multiparameter
Schur Q-functions (see also [8, Theorem 10.2])
Corollary 7.5. For any sequence of complex numbers a = (0, a1, a2, . . . ) and any integer vector
α = (α1, . . . , αl) with αi ∈ Z>0,
Q(a)
α =
∑
λ1,...,λl∈Z>0
(−1)
∑
λi−
∑
αieα1−λ1(a1, . . . , aα1−1) · · · eαl−λl(a1, . . . , aαl−1)Qλ.
Theorem 7.6. For any sequence of complex numbers a = (0, a1, a2, . . . ) and any integer vector
α = (α1, . . . , αl) with αi ∈ Z>0, multiparameter Schur Q-function Q
(a)
α is a solution of (5.1).
Proof. The constant polynomial 1 is obviously a solution of (5.1) in Bodd. By the vertex
operator presentation (3.4) and Corollary 7.5,
Q(a)
α =
∑
λ1,...,λl∈Z>0
Aλ1,α1 · · ·Aλl,αlϕλl · · ·ϕλ1 · 1
with An,k = (−1)n−kek−n(a1, . . . , ak−1). Hence,
Q(a)
α = Xαl ·Xα1 · 1,
where Xm =
∑
s>0
(−1)m−ses−m(a1, . . . , as−1)ϕs, and Corollary 6.2 proves the statement. �
Acknowledgements
The author would like to thank the referee for the thoughtful and careful review that helped to
improve the text of the paper.
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1 Introduction
2 Schur Q-functions
3 Action of neutral fermions on bosonic space B odd
4 Vertex operator form of formal distribution of neutral fermions
5 The neutral fermions bilinear identity
6 Commutation relation for the bilinear identity
7 Multiparameter Schur Q-functions are solutions of the BKP hierarchy
References
|
| id | nasplib_isofts_kiev_ua-123456789-210230 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:50Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Rozhkovskaya, N. 2025-12-04T13:03:58Z 2019 Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy / N. Rozhkovskaya // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 19 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05E05; 17B65; 17B69; 11C20 arXiv: 1805.06971 https://nasplib.isofts.kiev.ua/handle/123456789/210230 https://doi.org/10.3842/SIGMA.2019.065 We prove that multiparameter Schur Q-functions, which include as specializations factorial Schur Q-functions and classical Schur Q-functions, provide solutions of the BKP hierarchy. The author would like to thank the referee for the thoughtful and careful review that helped to improve the text of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy Article published earlier |
| spellingShingle | Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy Rozhkovskaya, N. |
| title | Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy |
| title_full | Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy |
| title_fullStr | Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy |
| title_full_unstemmed | Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy |
| title_short | Multiparameter Schur Q-Functions Are Solutions of the BKP Hierarchy |
| title_sort | multiparameter schur q-functions are solutions of the bkp hierarchy |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210230 |
| work_keys_str_mv | AT rozhkovskayan multiparameterschurqfunctionsaresolutionsofthebkphierarchy |