Virasoro Action on the -Functions

A formula for Schur -functions is presented, which describes the action of the Virasoro operators. For a strict partition, we prove a concise formula for ₋ₖ λ, where ₋ₖ ( ≥ 1) is the Virasoro operator.

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Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2021
Main Authors: Aokage, Kazuya, Shinkawa, Eriko, Yamada, Hiro-Fumi
Format: Article
Language:English
Published: Інститут математики НАН України 2021
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/211438
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Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:Virasoro Action on the -Functions. Kazuya Aokage, Eriko Shinkawa and Hiro-Fumi Yamada. SIGMA 17 (2021), 089, 12 pages

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
_version_ 1859975843349004288
author Aokage, Kazuya
Shinkawa, Eriko
Yamada, Hiro-Fumi
author_facet Aokage, Kazuya
Shinkawa, Eriko
Yamada, Hiro-Fumi
citation_txt Virasoro Action on the -Functions. Kazuya Aokage, Eriko Shinkawa and Hiro-Fumi Yamada. SIGMA 17 (2021), 089, 12 pages
collection DSpace DC
container_title Symmetry, Integrability and Geometry: Methods and Applications
description A formula for Schur -functions is presented, which describes the action of the Virasoro operators. For a strict partition, we prove a concise formula for ₋ₖ λ, where ₋ₖ ( ≥ 1) is the Virasoro operator.
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 089, 12 pages Virasoro Action on the Q-Functions Kazuya AOKAGE a, Eriko SHINKAWA b and Hiro-Fumi YAMADA c a) Department of Mathematics, National Institute of Technology, Ariake College, Fukuoka 836-8585, Japan E-mail: aokage@ariake-nct.ac.jp b) Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan E-mail: eriko.shinkawa.e8@tohoku.ac.jp c) Department of Mathematics, Kumamoto University, Kumamoto 860-8555, Japan E-mail: hfyamada@kumamoto-u.ac.jp Received June 10, 2021, in final form October 05, 2021; Published online October 08, 2021 https://doi.org/10.3842/SIGMA.2021.089 Abstract. 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−kQλ, where L−k (k ≥ 1) is the Virasoro operator. Key words: Q-functions; Virasoro operators 2020 Mathematics Subject Classification: 17B68; 05E10 To Minoru Wakimoto on his 80th birthday 1 Introduction The aim of this paper is to discuss Schur Q-functions in connection with a representation of the Virasoro algebra. Schur Q-functions are labelled by strict partitions and are defined as the Pfaffian of an alternating matrix. Let A = (aij)1≤i,j≤2m be an alternating 2m×2m matrix. The Pfaffian of A is Pf(A) := ∑ σ∈F2m (sgnσ)aσ(1)σ(2)aσ(3)σ(4) · · · aσ(2m−1)σ(2m), where F2m := {σ ∈ S2m;σ(1) < σ(3) < · · · < σ(2m− 1), σ(i) < σ(i+ 1) (i = 1, 3, . . . , 2m− 1)}. We see that |F2m| = (2m − 1)!!. The Laplace expansion of Pf(A) is as follows. For 1 ≤ i1 < · · · < i2ℓ ≤ 2m, let Ai1i2...i2ℓ be the 2ℓ × 2ℓ alternating matrix consisting of i1th row, i2th row, . . . , and i1th column, i2th column, . . . . Then Pf(A) = 2m∑ i=2 (−1)i Pf(A1i) Pf(A2...̂i...2m). Here î 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 LkQλ for k ≥ 1, where Lk denotes the Virasoro operator. As a continuation of [1] we give in the present paper a formula for L−kQλ. 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 mailto:aokage@ariake-nct.ac.jp mailto:eriko.shinkawa.e8@tohoku.ac.jp mailto:hfyamada@kumamoto-u.ac.jp https://doi.org/10.3842/SIGMA.2021.089 2 K. Aokage, E. Shinkawa and H.-F. Yamada 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. 2 Schur’s Q-functions A partition is an integer sequence λ = (λ1, λ2, . . . , λℓ), λ1 ≥ λ2 ≥ · · · ≥ λℓ > 0, whose size is |λ| = λ1 + λ2 + · · · + λℓ. The number of nonzero parts is the length of λ, denoted by ℓ(λ). Let SP(n) be the set of partitions of n into distinct parts. We call a λ ∈ SP(n) strict partition of n. Let V = C[tj ; j ≥ 1, odd]. This is decomposed as V = ⊕∞ n=0 V (n), where V (n) is the space of homogeneous polynomials of degree n, according to the counting deg tj = j. An inner product of V is defined by ⟨F,G⟩ = F ( 2∂̃ ) G(t)|t=0, where 2∂̃ = ( 2∂1, 2 3∂3, 2 5∂5, . . . ) with ∂j = ∂ ∂tj . Schur’s Q-functions are defined in our context as follows. Put ξ(t, u) = ∑ j≥1, odd tju j and define qn(t) ∈ V (n) by eξ(t,u) = ∞∑ n=0 qn(t)u n. For integers a, b with a > b > 0, define Qab(t) := qa(t)qb(t) + 2 b∑ i=1 (−1)iqa+i(t)qb−i(t), Qba(t) := −Qab(t). Finally, the Q-function labelled by the strict partition λ = (λ1, λ2, . . . , λ2m) (λ1 > λ2 > · · · > λ2m ≥ 0) is defined by Qλ(t) = Qλ1λ2...λ2m(t) = Pf (Qλiλj ) 1≤i,j≤2m . The Q-function Qλ(t) is homogeneous of degree |λ|. It is known that {Qλ(t); |λ| = n} forms an orthogonal basis for V (n), with respect to the above inner product. As Pfaffians, they satisfy the quadratic relations (cf. [2]): If ℓ(λ) is odd, Qλ1λ2...λℓ (t) = ℓ∑ i=1 (−1)i+1qλi (t)Q λ1...λ̂i...λℓ (t). If ℓ(λ) is even, Qλ1λ2...λℓ (t) = ℓ∑ i=2 (−1)iQλ1λi (t)Q λ2...λ̂i...λℓ (t). It is convenient to define Q-function Qα(t) for any non-negative integer sequence α = (α1, α2, . . . , αℓ). We adopt the following rule for permutations of indices: Virasoro Action on the Q-Functions 3 1. If α1, α2, . . . , αℓ are all distinct, then σ(α) is a strict partition for some permutation σ ∈ Sℓ, and Qα(t) = (sgnσ)Qσ(α)(t). 2. If αi = αj > 0 for some i ̸= j, then Qα(t) = 0. 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. For example, we have Q0,2,3,0,1(t) = −Q3,2,1(t). Detailed arguments are found in [2, Theo- rem 9.2]. Note that the above quadratic relations hold for Qα(t) with non-negative integer sequence α = (α1, α2, . . . , αℓ). We also agree that, for a > 0, Qa,−a(t) = (−1)a−1. Lemma 2.1. Let α = (α1, α2, . . . , αℓ) be a non-negative integer sequence, and x, y be non- negative integers. (1) If ℓ(α) is odd, Qαx = −qxQα − ℓ∑ i=1 (−1)iqαiQα1...α̂i...αℓx − ℓ∑ i=1 (−1)iQαixQα1...α̂i...αℓ , (2) If ℓ(α) is even, Qαx = −qxQα + ℓ∑ i=2 (−1)iQα1αiQα2...α̂i...αℓx + ℓ∑ i=2 (−1)iQα1αixQα2...α̂i...αℓ , (3) If ℓ(α) is odd, Qαxy = −QxyQα − ℓ∑ i=1 (−1)iqαiQα1...α̂i...αℓxy − ℓ∑ i=1 (−1)iQαixyQα1...α̂i...αℓ , (4) If ℓ(α) is even, Qαxy = −QxyQα + ℓ∑ i=2 (−1)iQα1αiQα2...α̂i...αℓxy + ℓ∑ i=2 (−1)iQα1αixyQα2...α̂i...αℓ . Proof. Let ℓ(α, x, y) be an even number. From the Pfaffian identity for Qαxy, the case (4) follows easily. The cases (3) and (2) follow from (4) by setting α1 = 0 and y = 0, respectively. Finally case (1) is obtained from case (3) by setting y = 0. ■ Next, we recall the boson-fermion correspondence for neutral free fermions ϕi (i ∈ Z) (cf. [3]). The Clifford algebra B is generated by free fermions ϕi (i ∈ Z) satisfying the anti-commutation relation: [ϕi, ϕj ]+ = (−1)iδi,−j . The vector space FB has a basis consisting of ϕi1ϕi2 · · ·ϕis |0⟩, i1 > i2 > · · · > is ≥ 0, where |0⟩ is the vacuum vector. The Clifford algebra B acts on FB by ϕi|0⟩ = 0, i < 0. For odd number n, we define the Hamiltonian by HB n = 1 2 ∑ i∈Z (−1)i−1ϕiϕ−n−i. 4 K. Aokage, E. Shinkawa and H.-F. Yamada The operators HB n (n ∈ Zodd) generate a Heisenberg algebra HB with [ HB n , HB m ] = n 2 δn,−m. It is known that FB is isomorphic to V : σB(|0⟩) = 1, σB ( HB n |0⟩ ) := ∂ ∂pn , σB ( HB −n|0⟩ ) := npn, n ≥ 1, odd. The map σB : HB −→ V is called the boson-fermion correspondence of type B. Moreover, for the basis of FB σB (ϕλ1ϕλ2 · · ·ϕλℓ |0⟩) = { 2− ℓ 2Qλ1λ2...λℓ if ℓ is even, 2− ℓ+1 2 Qλ1λ2...λℓ if ℓ is odd. In what follows, we denote σB (ϕλ1ϕλ2 · · ·ϕλℓ |0⟩) by ϕλ1ϕλ2 · · ·ϕλℓ |0⟩. Proposition 2.2. Let n = 2m. Then m−1∑ i=0 (2i+ 1)t2i+1(n− (2i+ 1))tn−(2i+1) = 2 m−1∑ i=0 (−1)i(m− i)Qn−i,i. Proof. First we rewrite the left-hand side of this equation by using power sum symmetric functions. m−1∑ i=0 (2i+ 1)t2i+1(n− (2i+ 1))tn−(2i+1) = 4 n∑ i≥1, odd pipn−i. For an odd number i, the operator HB i acts on FB. By the boson-fermion correspondence, the right hand side equals n∑ i≥1, odd ∑ j,k∈Z (−1)j+kϕjϕ−j+iϕkϕ−k+(n−i)|0⟩. Since free fermions ϕi (i < 0) act on vacuum vector |0⟩ as 0, the above summation becomes n∑ i≥1, odd ∑ −n+i≤j≤n 0≤k≤n−i (−1)j+kϕjϕ−j+iϕkϕ−k+(n−i)|0⟩ = n∑ i≥1, odd ∑ −n+i≤j<0 0≤k≤n−i (−1)j+kϕjϕ−j+iϕkϕ−k+(n−i)|0⟩ (2.1) + n∑ i≥1, odd ∑ 0≤j≤i 0≤k≤n−i (−1)j+kϕjϕ−j+iϕkϕ−k+(n−i)|0⟩ (2.2) + n∑ i≥1, odd ∑ i<j≤n 0≤k≤n−i (−1)j+kϕjϕ−j+iϕkϕ−k+(n−i)|0⟩. (2.3) Here it is verified that the part (2.2) equals 0. Next, we consider the parts (2.1) and (2.3). For the term (−1)j+kϕjϕ−j+iϕkϕ−k+(n−i), we only need to consider the cases that j or −j + i belongs to {−k,−(n− i) + k}. That is, (−1)j+kϕjϕ−j+iϕkϕ−k+(n−i) =  ϕ−kϕi+kϕkϕ−k+(n−i) if j = −k, −ϕi+kϕ−kϕkϕ−k+(n−i) if j = i+ k, −ϕ−(n−i)+kϕ−j+iϕkϕ−k+(n−i) if j = −(n− i) + k, ϕn−kϕ−j+iϕkϕ−k+(n−i) if j = n− k, Virasoro Action on the Q-Functions 5 = { (−1)k+1ϕi+kϕ−k+(n−i) if j = −k or j = i+ k, (−1)k+1ϕkϕ−k+n if j = −(n− i) + k or j = n− k. Hence we have n∑ i≥1, odd ∑ −n+i≤j≤n 0≤k≤n−i (−1)j+kϕjϕ−j+iϕkϕ−k+(n−i)|0⟩ = n∑ i≥1, odd ∑ 0≤k≤n−i (−1)k+1 (2ϕi+kϕ−k+(n−i) + 2ϕkϕ−k+n ) |0⟩ = 2 n∑ i≥1, odd ∑ 0≤k≤n−i (−1)k+1 (ϕi+kϕ−k+(n−i) + ϕkϕ−k+n ) |0⟩ = 2 m−1∑ i=0 (−1)i(n− 2i)ϕn−iϕi|0⟩ = 2 m−1∑ i=0 (−1)i(m− i)Qn−i,i. ■ 3 Reduced Fock representation of the Virasoro algebra For a positive odd integer j, put aj = √ 2∂j and a−j = j√ 2 tj so that they satisfy the Heisenberg relation as operators on V : [aj , ai] = jδj+i,0. For an integer k, put Lk = 1 2 ∑ j∈Zodd :a−jaj+2k: + 1 8 δk,0, where :ajai: = { ajai if j ≤ i, aiaj if j > i is the normal ordering. For example, L2 = 2∂1∂3 + ∑ j≥1, odd jtj∂j+4, L1 = ∂2 1 + ∑ j≥1, odd jtj∂j+2, L0 = ∑ j≥1 jtj∂j + 1 8 id, L−1 = 1 4 t21 + ∑ j≥3, odd jtj∂j−2, and L−2 = 3 2 t1t3 + ∑ j≥5, odd jtj∂j−4. More generally, it is verified, by Proposition 2.2, that L−k = 1 4 k−1∑ i=0 (2i+ 1)t2i+1(2k − (2i+ 1))t2k−(2i+1) + ∑ i≥2k+1, odd iti∂i−2k 6 K. Aokage, E. Shinkawa and H.-F. Yamada = 1 2 k−1∑ j=0 (−1)j(k − j)Q2k−j,j + ∑ j≥2k+1, odd jtj∂j−2k. It is known that the operators Lk on V satisfy the Virasoro relation: [Lk, Lℓ] = 2(k − ℓ)Lk+ℓ + k3 − k 3 δk+ℓ,0, k, ℓ ∈ Z. A representation of the Virasoro algebra L = ⊕k∈ZCℓk ⊕ Cz with central charge 1 is given by ℓk 7→ 1 2Lk, z 7→ 1, which we recall the reduced Fock representation. We have Lk · v ∈ V (n− 2k) for v ∈ V (n). The inner product ⟨ , ⟩ defined in Section 2 is contravariant: ⟨Lkv, w⟩ = ⟨v, L−kw⟩, v, w ∈ V. 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 exam- ple [5]. Proposition 3.1. L−1qn = (n+ 1)qn+2 + 1 2 Qn2, n ≥ 0. Proof. It is verified that L−1e ξ(t,u) = ( t21 4 + 3t3∂1 + 5t5∂3 + · · · ) eξ(t,u) = ( t21 4 + 3t3u+ 5t5u 3 + · · · ) e ∑ j≥1, odd tju j and ∂ ∂u eξ(t,u) = ∂ ∂u ( e ∑ j≥1, odd tju j ) = ( t1 + 3t3u 2 + 5t5u 4 + 7t7u 6 + · · · ) e ∑ j≥1, odd tju j . By the relations t1 = q1 and t1 2 = 2q2, we have uL−1e ξ(t,u) = ( 1 2 uq2 − q1 ) eξ(t,u) + ∂ ∂u eξ(t,u). Here ( 1 2 uq2 − q1 ) ∞∑ n=0 qnu n = 1 2 q2 ∞∑ n=0 qnu n+1 − q1 ∞∑ n=0 qnu n = u ( 1 2 q2 ∞∑ n=0 qnu n − q1 ∞∑ n=0 qn+1u n ) − q1, and ∂ ∂u ∞∑ n=0 qnu n = ∞∑ n=0 nqnu n−1 = u ∞∑ n=0 (n+ 2)qn+2u n + q1. Therefore we have L−1qn = 1 2 q2qn − q1qn+1 + (n+ 2)qn+2 = (n+ 1)qn+2 + 1 2 Qn2. ■ Virasoro Action on the Q-Functions 7 Similarly, we have Proposition 3.2. L−2qn = (n+ 2)qn+4 +Qn4 − 1 2 Qn3,1, n ≥ 0. Proof. It is verified that L−2e ξ(t,u) = ( 3 2 t1t3 + 5t5∂1 + 7t7∂3 + · · · ) eξ(t,u) = ( 3 2 t1t3 + 5t5u+ 7t7u 3 + · · · ) e ∑ j≥1, odd tju j and ∂ ∂u eξ(t,u) = ∂ ∂u ( e ∑ j≥1, odd tju j ) = ( t1 + 3t3u 2 + 5t5u 4 + 7t7u 6 + · · · ) e ∑ j≥1, odd tju j . Therefore u3L−2e ξ(t,u) = ( −t1 − 3t3u 2 + 3 2 t1t3u 3 ) eξ(t,u) + ∂ ∂u eξ(t,u). We have t3 = 1 3(q3 −Q2,1) and t1t3 = 1 3(2q4 −Q3,1). Therefore u3L−2e ξ(t,u) = ( −q1 − u2(q3 −Q2,1) + 1 2 u3(2q4 −Q3,1) ) eξ(t,u) + ∂ ∂u eξ(t,u). Here the first term equals −q1 ∞∑ n=0 qnu n − (q3 −Q2,1) ∞∑ n=0 qnu n+2 + 1 2 (2q4 −Q3,1) ∞∑ n=0 qnu n+3 = u3 ( −q1 ∞∑ n=0 qn+3u n − (q3 −Q2,1) ∞∑ n=0 qn+1u n + 1 2 (2q4 −Q3,1) ∞∑ n=0 qnu n ) + u3 ( −q1 ( u−3 + q1u −2 + q2u −1 ) − (q3 −Q2,1)u −1 ) and the second term equals ∞∑ n=0 nqnu n−1 = u3 ∞∑ n=0 (n+ 4)qn+4u n + u3 ( q1u −3 + 2q2u −2 + 3q3u −1 ) . Also it is easy to check that u3 ( −q1 ( u−3 + q1u −2 + q2u −1 ) − (q3 −Q2,1)u −1 ) + u3 ( q1u −3 + 2q2u −2 + 3q3u −1 ) = u2(2q3 − q1q2 +Q2,1) = 0. Hence L−2qn = −q1qn+3 − (q3 −Q2,1)qn+1 + 1 2 (2q4 −Q3,1)qn + (n+ 4)qn+4 = −q1qn+3 − (3q3 − q2q1)qn+1 + 1 2 (4q4 − q3q1)qn + (n+ 4)qn+4 = (n+ 2)qn+4 +Qn4 − 1 2 Qn3,1. ■ 8 K. Aokage, E. Shinkawa and H.-F. Yamada From Proposition 2.2, we obtain the following formula. L−k(vw) = (L−kv)w + v(L−kw)− 1 4 k−1∑ j=0 (2j + 1)t2j+1(2k − (2j + 1))t2k−(2j+1)vw = (L−kv)w + v(L−kw)− 1 2 k−1∑ j=0 (−1)j(k − j)Q2k−j,jvw for v, w ∈ V . In particular k = 1, 2, we see L−1(vw) = (L−1v)w + v(L−1w)− 1 2 q2vw, (3.1) L−2(vw) = (L−2v)w + v(L−2w)− 1 2 (2q4 −Q3,1)vw. (3.2) Proposition 3.3. Let α = (α1, α2, . . . , αℓ) be a positive integer sequence. Then (1) L−1Qα = ℓ∑ i=1 (αi + 1)Qα+2ϵi + 1 2 Qα,2, (2) L−2Qα = ℓ∑ i=1 (αi + 2)Qα+4ϵi +Qα,4 − 1 2 Qα,3,1. Proof. If αi = αj 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 ℓ(λ) is odd. By equation (3.1), L−1 ( ℓ∑ i=1 (−1)i+1qλi Q λ1...λ̂i...λℓ ) = ℓ∑ i=1 (−1)i+1 ( (L−1qλi )Q λ1...λ̂i...λℓ + qλi ( L−1Qλ1...λ̂i...λℓ ) − 1 2 q2qλi Q λ1...λ̂i...λℓ ) . (3.3) By induction hypothesis, and the first term and second term in the right hand side equal, respectively, ℓ∑ i=1 (−1)i+1 ( (λi + 1)qλi+2 + 1 2 Qλi2 ) Q λ1...λ̂i...λℓ , and ℓ∑ i=1 (−1)i+1qλi  ℓ∑ j=1, j ̸=i (λj + 1)Q λ1...λ̂i...λℓ+2ϵj + 1 2 Q λ1...λ̂i...λℓ2  . Hence the equation (3.3) reads ℓ∑ i=1 (−1)i+1 (λi + 1)qλi+2Qλ1...λ̂i...λℓ + qλi ℓ∑ j=1,j ̸=i (λj + 1)Q λ1...λ̂i...λℓ+2ϵj  + 1 2 ℓ∑ i=1 (−1)i+1(Qλi2Qλ1...λ̂i...λℓ + qλi Q λ1...λ̂i...λℓ2 − q2qλi Q λ1...λ̂i...λℓ ) = ℓ∑ i=1 (λi + 1)Qλ+2ϵi + 1 2 ℓ∑ i=1 (−1)i+1(Qλi2Qλ1...λ̂i...λℓ + qλi Q λ1...λ̂i...λℓ2 ) − 1 2 q2Qλ. Virasoro Action on the Q-Functions 9 By Lemma 2.1(1), the result follows. The case of even ℓ(λ) is similar. Next we prove (2) in Proposition 3.3. Let ℓ(λ) be odd. By equation (3.2), L−2 ( ℓ∑ i=1 (−1)i+1qλi Q λ1...λ̂i...λℓ ) (3.4) = ℓ∑ i=1 (−1)i+1 ( (L−2qλi )Q λ1...λ̂i...λℓ + qλi ( L−2Qλ1...λ̂i...λℓ ) − 1 2 (2q4 −Q3,1)qλi Q λ1...λ̂i...λℓ ) . By the induction hypothesis, the first and second terms in the right hand side are, respectively, ℓ∑ i=1 (−1)i+1 ( (λi + 2)qλi+4 +Qλi4 − 1 2 Qλi3,1 ) Q λ1...λ̂i...λℓ , and ℓ∑ i=1 (−1)i+1qλi  ℓ∑ j=1,j ̸=i (λj + 2)Q λ1...λ̂i...λℓ+4ϵj +Q λ1...λ̂i...λℓ4 − 1 2 Q λ1...λ̂i...λℓ3,1  . Hence the equation (3.4) reads ℓ∑ i=1 (−1)i+1 (λi + 2)qλi+4Qλ1...λ̂i...λℓ + qλi ℓ∑ j=1,j ̸=i (λj + 2)Q λ1...λ̂i...λℓ+4ϵj  + ℓ∑ i=1 (−1)i+1(Qλi4Qλ1...λ̂i...λℓ + qλi Q λ1...λ̂i...λℓ4 − q4qλi Q λ1...λ̂i...λℓ ) + 1 2 ℓ∑ i=1 (−1)i+1(−Qλi3,1Qλ1...λ̂i...λℓ − qλi Q λ1...λ̂i...λℓ3,1 +Q3,1qλi Q λ1...λ̂i...λℓ ) . Hence ℓ∑ i=1 (λi + 2)Qλ+4ϵi + ℓ∑ i=1 (−1)i+1(Qλi4Qλ1...λ̂i...λℓ + qλi Q λ1...λ̂i...λℓ4 ) − q4Qλ + 1 2 ℓ∑ i=1 (−1)i+1(−Qλi3,1Qλ1...λ̂i...λℓ − qλi Q λ1...λ̂i...λℓ3,1 ) + 1 2 Q3,1Qλ. The result follows immediately from Lemma 2.1(1) and (3). The case of even ℓ(λ) is similar. ■ Theorem 3.4. Let α = (α1, α2, . . . , αℓ) be a positive integer sequence. Then L−kQα = ℓ∑ i=1 (αi + k)Qα+2kϵi + 1 2 k−1∑ i=0 (−1)i(k − i)Qα,2k−i,i, k ≥ 1. Proof. Use induction on k. The cases k = 1, 2 are already shown in Proposition 3.3. Thanks to the Virasoro relations, it suffices to show [L−k, L−1]Qα = 2(−k + 1) ( ℓ∑ i=1 (αi + (k + 1))Qα+2(k+1)ϵi + 1 2 k∑ i=0 (−1)i((k + 1)− i)Qα,2(k+1)−i,i ) . 10 K. Aokage, E. Shinkawa and H.-F. Yamada Since L−kL−1Qα = ℓ∑ i=1 (αi + 1)L−kQα+2ϵi + 1 2 L−kQα2, and L−1L−kQα = ℓ∑ i=1 (αi + k)L−1Qα+2kϵi + 1 2 k−1∑ i=0 (−1)i(k − i)L−1Qα,2k−i,i, we have [L−k, L−1]Qα = ℓ∑ i=1 ( (αi + 1)L−kQα+2ϵi − (αi + k)L−1Qα+2kϵi ) + 1 2 ( L−kQα2 − k−1∑ i=0 (−1)i(k − i)L−1Qα,2k−i,i ) . We write down terms in the right hand side: ℓ∑ i=1 (αi + 1)L−kQα+2ϵi = ℓ∑ i,j=1,i ̸=j (αi + 1)(αj + k)Qα+2ϵi+2kϵj + ℓ∑ i=1 (αi + 1)(αi + k + 2)Qα+2(k+1)ϵi + ℓ∑ i=1 (αi + 1) 1 2 k−1∑ j=0 (−1)j(k − j)Qα+2ϵi,2k−j,j , ℓ∑ i=1 (αi + k)L−1Qα+2kϵi = ℓ∑ i=1 (αi + k)  ℓ∑ j=1,j ̸=i (αj + 1)Qα+2kϵi+2ϵj + (αi + 2k + 1)Qα+2(k+1)ϵi + 1 2 Qα+2kϵi,2  , L−kQα,2 = ℓ∑ i=1 (αi + k)Qα+2kϵi,2 + (2 + k)Qα,2+2k + 1 2 k−1∑ i=0 (−1)i(k − i)Qα,2,2k−i,i, k−1∑ i=0 (−1)i(k − i)L−1Qα,2k−i,i = k−1∑ i=0 (−1)i(k − i) ℓ∑ j=1 (αj + 1)Qα+2ϵj ,2k−i,i + k−1∑ i=0 (−1)i(k − i)(2k − i+ 1)Qα,2k−i+2,i + k−1∑ i=0 (−1)i(k − i)(i+ 1)Qα,2k−i,i+2 − 1 2 kQα,2k,2 + 1 2 k−1∑ i=1 (−1)i(k − i)Qα,2k−i,i,2. Summing up, we have( ℓ∑ i=1 (αi + 1)(αi + k + 2)− ℓ∑ i=1 (αi + k)(αi + 2k + 1) ) Qα+2(k+1)ϵi + 1 2 ( (2 + k)Qα,2+2k − k−1∑ i=0 (−1)i(k − i)(2k − i+ 1)Qα,2k−i+2,i + 1 2 k−1∑ i=0 (−1)i(k − i)Qα,2,2k−i,i + 1 2 kQα,2k,2 − 1 2 k−1∑ i=1 (−1)i(k − i)Qα,2k−i,i,2 Virasoro Action on the Q-Functions 11 − k−1∑ i=0 (−1)i(k − i)(i+ 1)Qα,2k−i,i+2 ) = 2(−k + 1) ℓ∑ i=1 (αi + (k + 1))Qα+2(k+1)ϵi + (−k + 1)(k + 1)Qα,2+2k − 1 2 ( k−1∑ i=1 (−1)i(k − i)(2k − i+ 1)Qα,2k−i+2,i + kQα,2k,2 + k−1∑ i=1 (−1)i(k − i)(i+ 1)Qα,2k−i,i+2 ) = 2(−k + 1) ℓ∑ i=1 (αi + (k + 1))Qα+2(k+1)ϵi + (−k + 1)(k + 1)Qα,2+2k + (−k + 1) k∑ i=1 (−1)i((k + 1)− i)Qα,2(k+1)−i,i. ■ Together with the previously proved formula LkQλ = 2m∑ i=1 (λi − k)Qλ−2kϵi for λ = (λ1, . . . , λ2m), k ≥ 1 [1, Theorem 2], Theorem 3.4 completely describes the reduced Fock representation of the Virasoro algebra. Consider the Lie subalgebra g = ∑ |k|≤1Clk which is isomorphic to sl(2,C). Let ESP be the set of the strict partitions whose parts are all even numbers, and let V even be the subspace of V spanned by the Qλ for λ ∈ ESP. 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 Qλ ( D̃ ) τ · τ = 0, λ ∈ SP\ESP coincides with those of the KdV hierarchy, where D̃ = ( D1, 1 3D3, 1 5D5, . . . ) is the Hirota differ- ential operator. Namely ( V even )⊥ is conjecturally the space of Hirota equations for the KdV hierarchy. For example, Q3,1 ( D̃ ) = 1 12 D4 1 − 1 3 D1D3 corresponds to the original KdV equation. Acknowledgements After completing the draft of the present paper we had a chance to see the preprint [4], where the same formula as ours is proved as a part of authors’ theory on BGW tau functions. Since the proof is slightly different, we decided to keep our proof in the present paper. We are grateful to the referees for informing [4] to us, as well as other useful comments which improved the paper. The funding was provided by KAKENHI (Grant No. 17K05180). 12 K. Aokage, E. Shinkawa and H.-F. Yamada References [1] Aokage K., Shinkawa E., Yamada H.F., Pfaffian identities and Virasoro operators, Lett. Math. Phys. 110 (2020), 1381–1389. [2] Hoffman P.N., Humphreys J.F., Projective representations of the symmetric groups. Q-functions and shifted tableaux, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992. [3] Jimbo M., Miwa T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), 943–1001. [4] Liu X., Yang C., Q-polynomial expansion for Brezin–Gross–Witten tau-function, arXiv:2104.01357. [5] Wakimoto M., Yamada H.-F., The Fock representations of the Virasoro algebra and the Hirota equations of the modified KP hierarchies, Hiroshima Math. J. 16 (1986), 427–441. [6] Yamada H.-F., Reduced Fock representation of the Virasoro algebra, in Proceedings of the 35th Symposium on Algebraic Combinatorics, 2018, 38–45, available at https://hnozaki.jimdofree.com/ proceedings-symp-alg-comb/no-35/. https://doi.org/10.1007/s11005-020-01265-1 https://doi.org/10.2977/prims/1195182017 https://arxiv.org/abs/2104.01357 https://doi.org/10.32917/hmj/1206130440 https://hnozaki.jimdofree.com/proceedings-symp-alg-comb/no-35/ https://hnozaki.jimdofree.com/proceedings-symp-alg-comb/no-35/ 1 Introduction 2 Schur's Q-functions 3 Reduced Fock representation of the Virasoro algebra References
id nasplib_isofts_kiev_ua-123456789-211438
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1815-0659
language English
last_indexed 2026-03-18T05:19:50Z
publishDate 2021
publisher Інститут математики НАН України
record_format dspace
spelling Aokage, Kazuya
Shinkawa, Eriko
Yamada, Hiro-Fumi
2026-01-02T08:33:39Z
2021
Virasoro Action on the -Functions. Kazuya Aokage, Eriko Shinkawa and Hiro-Fumi Yamada. SIGMA 17 (2021), 089, 12 pages
1815-0659
2020 Mathematics Subject Classification: 17B68; 05E10
arXiv:2106.04773
https://nasplib.isofts.kiev.ua/handle/123456789/211438
https://doi.org/10.3842/SIGMA.2021.089
A formula for Schur -functions is presented, which describes the action of the Virasoro operators. For a strict partition, we prove a concise formula for ₋ₖ λ, where ₋ₖ ( ≥ 1) is the Virasoro operator.
After completing the draft of the present paper, we had a chance to see the preprint [4], where the same formula as ours is proved as a part of the authors’ theory on BGW tau functions. Since the proof is slightly different, we decided to keep our proof in the present paper. We are grateful to the referees for informing us [4], as well as for other useful comments that improved the paper. The funding was provided by KAKENHI (Grant No. 17K05180).
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Virasoro Action on the -Functions
Article
published earlier
spellingShingle Virasoro Action on the -Functions
Aokage, Kazuya
Shinkawa, Eriko
Yamada, Hiro-Fumi
title Virasoro Action on the -Functions
title_full Virasoro Action on the -Functions
title_fullStr Virasoro Action on the -Functions
title_full_unstemmed Virasoro Action on the -Functions
title_short Virasoro Action on the -Functions
title_sort virasoro action on the -functions
url https://nasplib.isofts.kiev.ua/handle/123456789/211438
work_keys_str_mv AT aokagekazuya virasoroactiononthefunctions
AT shinkawaeriko virasoroactiononthefunctions
AT yamadahirofumi virasoroactiononthefunctions