Virasoro Action on the $Q$-Functions

A formula for Schur $Q$-functions is presented which describes the action of the Virasoro operators. For a strict partition, we prove a concise formula for $L_{-k}Q_{\lambda}$, where $L_{-k}$ $(k\geq 1)$ is the Virasoro operator.

We see that |F 2m | = (2m − 1)!!. The Laplace expansion of Pf(A) is as follows. For 1 ≤ i 1 < · · · < i 2 ≤ 2m, let A i 1 i 2 ...i 2 be the 2 × 2 alternating matrix consisting of i 1 th row, i 2 th row, . . . , and i 1 th column, i 2 th column, . . Here i means the omission of i. We will utilize this quadratic relation to derive formulas for Q-functions.
Our previous paper [1] gives a formula of L k Q λ for k ≥ 1, where L k denotes the Virasoro operator. As a continuation of [1] we give in the present paper a formula for L −k Q λ . Section 2 is a review of Q-functions containing some identities which do not seem to be obviously derived from Pfaffian identities. In Section 3 we first recall the reduced Fock representation of the Virasoro algebra on the space of the Q-functions. Then the main result is given. Proofs consist of direct, simple calculations.
The Virasoro representations of this paper may be applied to, for example, the Kontsevich matrix models by certain rescaling. However we will not discuss here any relationship. Our motivation is to clarify the representation theoretical nature of the Hirota equations for certain soliton type hierarchies. In the final section we will give a conjecture on the Hirota equations for the KdV hierarchy.
3. Using permutations, 0's should be moved in the tail of α, keeping 0's order. After such permutation, all 0's should be deleted.
Next, we recall the boson-fermion correspondence for neutral free fermions φ i (i ∈ Z) (cf. [3]). The Clifford algebra B is generated by free fermions φ i (i ∈ Z) satisfying the anti-commutation relation: The vector space F B has a basis consisting of is the vacuum vector. The Clifford algebra B acts on F B by φ i |0 = 0, i < 0. For odd number n, we define the Hamiltonian by The operators H B n (n ∈ Z odd ) generate a Heisenberg algebra H B with H B n , H B m = n 2 δ n,−m . It is known that F B is isomorphic to V : In what follows, we denote Proof . First we rewrite the left-hand side of this equation by using power sum symmetric functions.
For an odd number i, the operator H B i acts on F B . By the boson-fermion correspondence, the right hand side equals n i≥1, odd j,k∈Z Since free fermions φ i (i < 0) act on vacuum vector |0 as 0, the above summation becomes n i≥1, odd −n+i≤j≤n Here it is verified that the part (2.2) equals 0. Next, we consider the parts (2.1) and (2.3).
Hence we have n i≥1, odd −n+i≤j≤n

Reduced Fock representation of the Virasoro algebra
For a positive odd integer j, put a j = √ 2∂ j and a −j = j √ 2 t j so that they satisfy the Heisenberg relation as operators on V : [a j , a i ] = jδ j+i,0 .
For an integer k, put is the normal ordering. For example, More generally, it is verified, by Proposition 2.2, that It is known that the operators L k on V satisfy the Virasoro relation: A representation of the Virasoro algebra L = ⊕ k∈Z C k ⊕ Cz with central charge 1 is given by k → 1 2 L k , z → 1, which we recall the reduced Fock representation. We have L k · v ∈ V (n − 2k) for v ∈ V (n). The inner product , defined in Section 2 is contravariant: Therefore the reduced Fock representation is infinitesimally unitary. The singular vectors are discussed in [6]. For the non-reduced Fock representation of the Virasoro algebra, see for example [5].
for v, w ∈ V . In particular k = 1, 2, we see Proposition 3.3. Let α = (α 1 , α 2 , . . . , α ) be a positive integer sequence. Then Proof . If α i = α j for some i = j, then the equations hold as 0 = 0. Therefore, taking the sign (±1) into account, it suffices to prove the equations for the case α = λ is a strict partition. Use induction on the length of λ. First we see (1) for the case (λ) is odd. By equation (3.1), By induction hypothesis, and the first term and second term in the right hand side equal, respectively,

Hence the equation (3.3) reads
By Lemma 2.1(1), the result follows. The case of even (λ) is similar. Next we prove (2) in Proposition 3.3. Let (λ) be odd. By equation (3.2), By the induction hypothesis, the first and second terms in the right hand side are, respectively,

Hence the equation (3.4) reads
The result follows immediately from Lemma 2.1 (1) and (3). The case of even (λ) is similar.
Corollary 3.5. The space V even is invariant under the action of g.
We are interested in the space V even because of the following conjecture: the set of Hirota bilinear equations Q λ D τ · τ = 0, λ ∈ SP\ESP coincides with those of the KdV hierarchy, where D = D 1 , 1 3 D 3 , 1 5 D 5 , . . . is the Hirota differential operator. Namely V even ⊥ is conjecturally the space of Hirota equations for the KdV hierarchy. For example, corresponds to the original KdV equation.