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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| 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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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 |