Multiparameter Schur Q-functions are solutions of 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 BKP hierarchy

Multiparameter Schur Q-functions Q (a) λ were introduced and studied combinatorially in [9]. These symmetric functions are interpolation analogues of the classical Schur Q-functions depending on a sequence of complex valued parameters a = (a 0 , a 1 , . . . ). Definition of multiparameter Schur Q-functions is reproduced in (7.1). Classical Schur Q-functions correspond to a = (0, 0, 0, . . . ), and with evaluation a = (0, 1, 2, 3, . . . ) multiparameter Schur Q-functions are called factorial Schur Qfunctions. These families of symmetric functions proved to be useful in study of a number of questions of representation theory and algebraic geometry. Here are few examples.
The authors of [1], [15] described Capelli polynomials of 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 factorial Schur Q-functions.
In [6], [8] the equivariant cohomology of a Lagrangian Grassmannian of a symplectic or orthogonal 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 symmetric functions. Further in [7] 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.
In [5] the center of the twisted version of Khovanov's Heisenberg category is identified with the algebra generated by classical Schur Q-functions (denoted as B odd 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 BKP hierarchy. The origin for this phenomena lies in the fact proved in [13] that generating functions of multiparameter Schur Q-functions and of classical Schur Q-functions coincide.
While 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 BKP hierarchy in terms of generating functions of symmetric functions with 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 [14], 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 KP and BKP integrable systems solely in terms of symmetric functions can be found e.g. in [10]. The authors of [10] start with the bilinear identities in integral form, then, using the Cauchy type orthogonality properties of symmetric functions (c.f. [14] Chapter III (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, elementary 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 adjoint operators and deduce vertex operator form of the formal distribution of neutral fermions. In Section 5 we review all the steps of recovering BKP hierarchy of partial differential equations 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 BKP hierarchy (which recovers the result of [17]). In Section 7 we introduce multiparameter Schur Q-funsctions, and using the observation of Section 6, we show that Q (a) λ are also solutions of BKP hierarchy.

Schur Q-functions
Let B be the ring of symmetric functions in variables (x 1 , x 2 , . . . ). Consider the families of the following symmetric functions: It is well-known ( [14] Chapter I.2), that each of these families generate B as a polynomial ring: Combine the families h k , e k , p k into generating functions The following facts are well-known ( [14] Chapter I.2). Lemma 2.1.
We introduce one more family of symmetric functions {Q k = Q k (x 1 , x 2 , . . . )} with (k = 0, 1, . . . ) as the coefficients of generating function Then from Lemma 2.1 and (2.1) we get immediately following relations.
Note that Q(u)Q(−u) = 1, which implies that Q r with even r can be expressed algebraically through Q r with odd r: More generally, Schur Q-functions Q λ labeled by strict partitions are defined as a specialization of Hall-Littlewood polynomials ( [14] Chapter III.2).
where it is understood that and Q(u) is given by (2.1) ( [14] Chapter III, (8.8)). Schur Q-polynomials have a stabilization property, hence, one can omit the number N of variables x ′ i s as long as it is not less than the length of the partition λ and consider Q λ as functions of infinitely many variables (x 1 , x 2 . . . ).

Action of neutral fermions on bosonic space B odd
Consider the subaglebra B odd of B generated by odd ordinary Schur Q-functions: It is known that B odd is also a polynomial algebra in odd power sums B odd = C[p 1 , p 3 , . . . ] and that Schur Q-functions Q λ labeled by strict partitions constitute a linear basis of B odd ( [14] Chapter III.8, (8.9)).
Define operators {ϕ k } k∈Z acting on the coefficients of generating functions Q(u 1 , . . . , u l ) by the rule Observe that from (2.3) Since coefficients of the expansion of Q(u 1 , . . . , u l ) in powers of u 1 , . . . , u l include Schur Q-functions Q λ , and the later form a linear basis of B odd , one can show that (3.1) provides the action of well-defined operators {ϕ k } k∈Z on B odd :

Vertex operator form of formal distribution of neutral fermions
It will be convenient for us to consider B odd as the subring of the ring of symmetric functions B. This allows us to recover the well-known 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 [14] Chapter I.
The ring of symmetric functions B possesses a bilinear form (·, ·) ( [14] Chapter I (4.5)) defined on the linear basis of monomials of power sums labeled by partitions λ and µ as where z λ = i m i m i ! and m i = m i (λ) is the number of parts of λ equal to i.
We will use this form and its restriction to B odd to define adjoint operators 1 of the multiplication operators. By definition, given an element f ∈ B, the operator f ⊥ adjoint to the operator of multiplication by f is given by the rule Chapter I. 5 Example 3 of [14] contains the following statement. Consider a symmetric function f = f (p 1 , p 2 , . . . ) expressed as a polynomial in power sums p i . Then adjoint operator on B to the multiplication operator by f is given by In particular p ⊥ n = n∂/∂p n . Combine the corresponding adjoint operators of the families h k , e k , p k and Q k into generating functions Then (4.1) immediately implies the following relations.
The proof of next lemma is outlined in [14] Chapter I.5 Example 29. 1 Traditionally, one uses rescaled form on B odd defined as (p λ , p µ ) = 2 −l(λ) z λ δ λ,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 · ∂/∂p n (see [14] Chapter III.8. Example 11) Lemma 4.2. The following commutation relations on generating functions of multiplication and adjoint operators acting on B hold.
Proof. For the first and second one we use that Q(u) = E(u)H(u). Observe that In other words, since Q k does not depend on even power sums p 2r , we can add terms ∂/∂p 2r in the sum under the exponent when applying to elements of B odd : We arrive at the vertex operator form of formal distribution of neutral fermions.
Since coefficients of Q(u 1 , . . . , u l ) contain linear basis of B odd , the equality (4.2) follows.

Neutral fermions bilinear identity
In this section we use the vertex operator form (4.2) of Φ(u) to convert neutral fermions bilinear identity into BKP hierarchy of partial differential equations. This is a well-known procedure that we provide here in details for the convenience of the reader. Then a simple observation why Schur Q-functions constitute solutions of neutral fermions bilinear identity, and hence of BKP hierarchy, follows.
The following lemma allows one to rewrite bilinear identity (5.1) in terms of Hirota derivatives.
Thus, we can write in terms of Hirota derivatives In order to compute Res u=0 1 u Φ(u)τ ⊗ Φ(−u)τ , which is just the coefficient of u 0 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 [14] Chapter I.
Then the following statements hold: 1 24 X 4 1 . By Lemma 2.1, when X variables in these formulas are interpreted as normalized power sums X k = p k /k, S k 's are identified with complete symmetric functions h k 's.
Using the statement of Proposition 5.1, we can write the coefficient of u 0 of (5.4) as Note that S 0 = 1 and exp n∈N odd y n D n τ · τ = τ (x − y) · τ (x + y), hence we can rewrite (5.5) as the term y 2 3 appears in S 3 (ỹ) S 3 (D) × y 3 D 3 and in S 6 (ỹ) S 6 (D) × 1. Using the expansions of Example 5.2, the coefficient of which provides the Hirota bilinear form of the BKP equation that gives the name to the hierarchy: (D 6 1 − 5D 1 D 3 − 5D 2 3 + 9D 1 D 5 )τ · τ = 0. Remark 5.1. Writing the residue (5.2) as a contour integral, one gets BKP in its integral form:

Commutation relation for the bilinear identity.
Our goal is to show that multiparameter Schur Q-functions are solutions of neutral fermions bilinear identity (5.1), thus they provide solutions of BKP hierarchy.
Let X = n>0 A n ϕ n for some A n ∈ C. From (3.3) one gets X 2 = 0.
We use that X 2 = 0, and since both m and n in the last sum are always positive, the last term is also zero.
Corollary 6.1. Let τ ∈ B odd be a solution of (5.1), and let X = n>0 A n ϕ n with A n ∈ C. Then τ ′ = Xτ is also a solution of (5.1).
Vertex operator presentation (3.4) of Schur Q-functions and Corollary 6.1 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 [17] and easily extends to more general case of multiparameter Schur Q-functions defined in the next section.