Principal subspace for the bosonic vertex operator ϕ2m(z) and Jack polynomials

Advances in Mathematics 206 (2006) 307–328 www.elsevier.com/locate/aim

Principal subspace for the bosonic vertex operator φ√2m(z) and Jack polynomials B. Feigin a,b , E. Feigin b,c,∗ a Landau Institute for Theoretical Physics, Chernogolovka 142432, Russia b Independent University of Moscow, Bol’shoi Vlas’evskii, 11, Moscow, Russia c Moscow State University, Mech-Math Faculty, Department of Higher Algebra,

Leninskie gori, 1, Moscow, Russia Received 27 July 2004; accepted 2 September 2005 Available online 13 February 2006 Communicated by Pavel Etingof

Abstract

 Let φ√2m (z) = n∈Z an z−n−m , m ∈ N, be a bosonic vertex operator and L be some irreducible representation of the vertex algebra A(m) associated with the one-dimensional lattice Zl, l, l = 2m. Fix some extremal vector v ∈ L. We study the principal subspace C[ai ]i∈Z · v and its finitization C[ai ]i>N · v. We construct their bases and find characters. In the case of finitization, the basis is given in terms of Jack polynomials. © 2005 Elsevier Inc. All rights reserved. Keywords: Vertex operators; Jack polynomials

0. Introduction 2 , and v2n ∈ L0,1 , Let L0,1 and L1,1 be irreducible representations of the Lie algebra sl v2n+1 ∈ L1,1 the extremal vectors  (for example, v0 is the vacuum vector). For x ∈ sl2 , we consider the current x(z) = i∈Z xi z−i−1 (we use the notation xi = x ⊗ t i ). Let e, h, f * Corresponding author.

E-mail addresses: [email protected] (B. Feigin), [email protected] (E. Feigin). 0001-8708/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2005.09.001

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be the standard basis of sl2 . Then ei vp = 0 for i  p. We consider the principal subspace Vp = C[ep−1 , ep−2 , . . .] · vp and list some of its properties (see [2,4,5,7]). (1) Vp  C[ep−1 , ep−2 , . . .]/Ip , where Ip is the ideal generated by coefficients of the series (ep−1 + zep−2 + z2 ep−3 + · · ·)2 . (2) The elements ei1 · · · eik v,

i1 < p, iα − iα+1  2, k = 0, 1, . . . ,

form a basis of Vp . Using this basis, one can write a formula for the character of Vp and construct the semi-infinite basis of L0,1 and L1,1 . (3) Let Vp (n) → Vp be the finitization of V (n), Vp (n) = C[ep−1 , . . . , ep−n ] · vp . Then dim Vp (n) = 2n . We recall that the current e(z) can be considered as the bosonic vertex operator φ√2 (z). In this paper, we generalize (1)–(3) to the case of φ√2m (z) for an arbitrary m ∈ N. (This generalization is mentioned in [6]. We also note that the case of the fermionic vertex operator φ√3 (z) is considered in [13].) algebra associated with a one-dimensional lattice Zl, Let A(m) be the lattice vertex  l, l = 2m, and φ√2m (z) = n∈Z an z−n−m be the corresponding bosonic vertex operator. Let L(m),i be the set of irreducible representations of A(m) (see [3]). They can be described as follows:  L(m),i = H 2nm+i , 0  i  2m − 1. √ n∈Z

2m

For any c ∈ C, Hc is a Fock module over the Heisenberg algebra H with the highest weight c. Note that the Fourier coefficients ai of φ√2m (z) act from Hc to Hc+√2m . Recall the action of the Virasoro algebra on L(m),i (hα are generators of H ), Ln =

1  m−1 : hα hβ : − √ (n + 1)hn , 2 2m α+β=n

√ where the factor −(m √ − 1)/ 2m is fixed by the condition [Ln , a0 ] = 0. We introduce a notation vp = |−p/ 2m , p ∈ Z, for the extremal vectors of L(m),i . Note that ai vp = 0 for i  p. Following [8], we define the principal subspace V(m),p = C[ap−1 , ap−2 , . . .] · vp . We first describe the ideal I(m),p such that V(m),p  C[ap−1 , ap−2 , . . .]/I(m),p . Namely, I(m),p is generated by the coefficients of the series  (i) 2 ap−1 + ap−2 z + ap−3 z2 + · · · ,

0  i < m,

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where the superscript (i) stands for the ith derivative. Using the functional realization (see ∗ , we obtain its character, construct the fermionic realization of [8]) of the dual space V(m),p V(m),p , and find its monomial basis. Namely, we prove that the elements ai 1 · · · ai k v p ,

i1  p − 1, iα − iα+1  2m, k = 0, 1, . . . ,

form a basis of V(m),p . We also show that this basis is compatible with the operation of taking coinvariants. This means that the images of the vectors ai1 · · · aik vp with iα − iα+1  2m,

i1  p − 1, ik > N,

form a basis of the quotient V(m),p /span{ai V(m),p , i  N }. We next consider the subspace V(m),p (n) → V(m),p , (m)

and define numbers Fi (m)

Fi

= i + 1,

V(m),p (n) = C[ap−1 , ap−2 , . . . , ap−n ] · vp

as (m)

i = 0, 1, . . . , m − 1,

(m)

Fi+m = Fi

(m)

+ Fi+m−1 ,

i  0.

For example, Fi(1) = 2i and Fi(2) are the Fibonacci numbers. We prove that the dimension (m) of V(m),p (n) is equal to Fn . The proof is based on the construction of a basis of V(m),p (n) 2 case). in terms of Jack polynomials Jλ (m; x) (which reduce to Schur polynomials in the sl We recall that any Fock module Hc can be considered as the space of polynomials in infinitely many variables via the identification (hi with i < 0 are generating operators)  hi1 · · · hik |c →

m 2

k p−i1 · · · p−ik ,

iα < 0,

where |c is the highest-weight vector of Hc and pj is the j th power sum. Combining the results in [12,14] (see also [1]), we obtain that a0k vp = C · J((p−(k−1)m−1)k ) (m; x),

(1)

where C is some nonvanishing constant and ((p − (k − 1)m − 1)k ) is the partition   p − (k − 1)m − 1, . . . , p − (k − 1)m − 1 .   k √ √ −p/ 2m +k 2m (n) V(m),p

be the intersection V(m),p (n) ∩ H−p/√2m +k √2m . We For p ∈ Z, let note that V(m),p is independent of p (up to a shift of the Fourier coefficients an ). Therefore, the spaces V(m),n (p) and V(m),p (p) are isomorphic. Denote the latter space simply by V(m) (p). We show that √−p 2m

V(m)

√ +k 2m

(p) = C[h1 , h2 , . . .] · a0k vp .

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In addition, C[h1 , h2 , . . .] · J((p−m(k−1)−1)k ) (m; x) is the linear span of Jack polynomials Jλ (m; x) with the Young diagrams λ being subdiagrams of ((p − m(k − 1) − 1)k ). This (combined with (1)) gives the dimension and character of V(m) (p). We recall that in [4], the spaces V(1) (n) were obtained via the fusion procedure. Let Z = (z1 , . . . , zn ) ∈ Cn , zi = zj . Consider the ring R = C[y1 , . . . , yn ]/yi2 1in and fix  its generators ai = nj=1 yj zji , i = 0, . . . , n − 1. We have R  C[a0 , . . . , an−1 ]/IZ (n), where IZ (n) is some ideal. Then the following is true:  V(1) (n)  C[a0 , . . . , an−1 ] lim IZ (n). Z→0

Conjecturally, an analogous construction exists for general m. Namely, let R = C[y1 , . . . , yn ]/yi yj |i−j |
The dimension of this ring is obviously equal to Fn . We discuss the fusion conjecture for the general m in the last section. 2 -representations L0,1 and L1,1 . As mentioned above, L(m),i generalize the irreducible sl Using the semi-infinite construction of L(m),i (see [8] for the case m = 1), we obtain bases of these spaces. This paper is organized as follows. In Section 1, we consider the quotient of C[ξ0 , ξ1 , . . .] by the ideal generated by coefficients of the series ((ξ0 + ξ1 z + · · ·)(k) )2 , k = 0, 1, . . . , m − 1. We find a basis of this quotient and a basis of the space of coinvariants. In Section 2, we recall the definition and the main properties of lattice vertex operator algebras. In Section 3, we study the principal subspaces V(m),p and finitizations V(m) (p). We find relations between Fourier coefficients of φ√2m (z) and construct bases of V(m),p and V(m) (p). Section 4 is devoted to the computation of the character and dimension of V(m) (p). Section 5 contains the construction of a semi-infinite basis for L(m),i . In the last section, we describe the fusion construction for V(m) (p). 1. Commutative algebras Let ξ0 , ξ1 , . . . be some commuting variables. We consider the algebra 2   A(m) = C[ξ0 , ξ1 , . . .] ξ(z)(i) 0im−1 ,

(2)

 j (i) its ith derivative, and the ideal of relations (the rightwhere ξ(z) = ∞ j =0 ξj z , ξ(z) hand side of (2)) is generated by the coefficients of (ξ(z)(i) )2 , i = 0, 1, . . . , m − 1. The algebra A(m) is bi-graded: A(m) =

 k,s0

 Ak,s (m) ,

Ak,s (m)

= span ξi1 · · · ξik ,

k  α=1

 iα = s .

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We set ch(A(m) ) =



k,s0 z

Lemma 1.1. ch(A(m) ) =

kqs

311

dim Ak,s (m) .



q mk(k−1) k k0 z (1−q)···(1−q k ) .

∗ Proof. The dual space (Ak,s (m) ) can be identified with a subspace of symmetric polynomials ∗ f (z1 , . . . , zk ) of degree s. Namely, for θ ∈ (Ak,s (m) ) , let



fθ (z1 , . . . , zk ) =

z1i1 · · · zkik θ (ξi1 · · · ξik ).

i1 ,...,ik 0 ∗ We show that f = fθ for some θ ∈ (Ak,s (m) ) if and only if



f (z1 , . . . , zk ) =

(zi − zj )2m g(z1 , . . . , zk )

(3)

1i
for some symmetric g. In fact, by definition,   fθ (z1 , . . . , zk ) = θ ξ(z1 ) · · · ξ(zk ) . Therefore, for any 1  i < j  k and 0  l  m − 1, 

∂ ∂ ∂zi ∂zj

l

fθ

zi =zj =z

  = θ · · · ξ(z)(l) · · · ξ(z)(l) · · · = 0.

This shows that fθ is divisible by (zi − zj )2m . Now suppose (3) holds. Then there exists a unique θ such that   fθ (z1 , . . . , zk ) = θ ξ(z1 ) · · · ξ(zk ) . Because of condition (3), θ vanishes on the ideal generated by the coefficients of the series ∗ ξ(z)(l) , 0  l  m − 1. Therefore, θ ∈ (Ak,s (m) ) . 2 We now construct a monomial basis of A(m) . For this, we embed A(m) into the algebra F generated by anticommuting variables ψs (i),  s = 1, . . . , 2m, i ∈ N ∪ {0}: ψs (i)ψt (j ) = −ψt (j )ψs (i). Consider the currents ψs (z) = i0 ψs (i)zi . Let ξ˜i ∈ F , i  0, be elements defined by ξ˜ (z) =



ξ˜i zi = ψ1 (z) · · · ψ2m (z).

i0

Lemma 1.2. The algebra generated by ξ˜i , i  0, is isomorphic to A(m) via the identification ξ˜i → ξi .

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Proof. First, it is easy to verify that (ξ˜ (z)(i) )2 = 0 for i = 0, . . . , m − 1. We need to prove that these are the defining relations. For this, it suffices to find a set of linearly independent monomials in the variables ξ˜0 , ξ˜1 , . . . whose character (with respect to the grading degz ξ˜i = 1, degq ξ˜i = i) is equal to the character of A(m) . We consider the set of admissible monomials ξ˜i1 · · · ξ˜ik ,

0  i1 , iα+1 − iα  2m, α = 1, . . . , k − 1,

and prove that they are linearly independent. In fact, let some linear combination of the admissible monomials vanish:  (4) αi1 ,...,ik ξ˜i1 · · · ξ˜ik = 0. Pick the monomial ξ˜i 0 · · · ξ˜i 0 with a nonzero αi 0 ,...,i 0 such that for any other k-tuple k k 1 1 (i1 , . . . , ik ) with αi1 ,...,ik = 0, the following is true: there exists l < k such that i10 = i1 , . . . , il0 = il ,

0 il+1 > il+1 .

We claim that ξ˜i 0 · · · ξ˜i 0 cannot be written as a sum of the other monomials in (4). k 1 In fact, ξ˜i 0 · · · ξ˜i 0 = 1

k

k 



   s  . ψ1 β1s · · · ψ2m β2m

(5)

s =i 0 s=1 β1s +···+β2m s

The right-hand side of (5) contains the term k  s=1

 ψ1

is0 2m

 0   0 



is + 1 is + 2m − 1 ψ2 · · · ψ2m , 2m 2m

(6)

which is nonzero because of the condition iα+1 − iα  2m ([x] is the maximum integer not exceeding x). By definition of (i10 , . . . , ik0 ), the term (6) appears in sum (4) only in the product ξ˜i 0 · · · ξ˜i 0 . Therefore (4) is impossible. k 1 To finish the proof, we note that the character of admissible monomials coincides with the character of A(m) . 2 Corollary 1.1. ξi1 · · · ξik , iα+1 − iα  2m, form a basis of A(m) . Proposition 1.1. The images of the monomials ξi1 · · · ξik , iα+1 − iα  2m, ik  N , in the quotient A(m) /span{ξi A(m) , i > N} form a basis of this quotient. Proof. Note that the term (6) with ik0  N cannot appear in the sum of the form ξN +1 x1 + ξN +2 x2 + · · · + ξN +p xp ,

xi ∈ A(m) .

(7)

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In fact, replacing ξi with ξ˜i , we obtain that (7) is a linear combination of the products of fermions, 2m 

    ψs β1s · · · ψs βks

s=1

with the property that there exists a map σ : {1, . . . , 2m} → {1, . . . , k} such that βσ1 (1) + · · · + βσ2m (2m) > N . But the latter is not true for (6). Therefore, the sum (7) does not contain monomial (6). We obtain that the images of the monomials ξi1 · · · ξik , iα+1 − iα  2m, ik  N , in the quotient A(m) /span{ξi A(m) , i > N} are linearly independent. The proposition now follows from Corollary 1.1. 2

2. One-dimensional lattice vertex operator algebras In this section, we recall the definition of one-dimensional lattice vertex operator algebras (VOA for short) and collect their main properties. The main references are [3,9,10]. A lattice VOA is determined by a lattice Γ → Rn and a scalar product ·,· on Rn taking integer values on vectors from Γ . Such an algebra is generated by vertex operators V (x, z), x ∈ Γ . In what follows, we concentrate on the case of a positive definite scalar product. Let n = 1 and let α be a basis vector of Γ , α, α = m, m ∈ N. For m = 1, 2, 3, . . . , let Am denote the corresponding VOA. Let + (z) = V (α, z) φ√ m

− and φ√ (z) = V (−α, z). m

± ± ± ± (z) φ√ (w) = φ√ (w) φ√ (z)) for even m and fermionic These fields are bosonic (φ√ m m m m

± ± ± ± (φ√ (z) φ√ (w) = −φ√ (w) φ√ (z)) otherwise. They satisfy the relation m m m m ± ± ± ± (z) φ√ (w) = (z − w)m : φ√ (z) φ√ (w) : , φ√ m m m m

where : : denotes the normal ordering. We note that the algebra Am for m = 1 can be considered as a infinite-dimensional 2 algebra at level k = 1; and for m = 3, the N = 2 Clifford algebra; for m = 2, the sl superconformal algebra. We recall some facts about representations of lattice VOAs. The category of representations of a lattice VOA is semisimple, i.e., any representation decomposes into a direct sum of irreducible ones. The irreducible representations are enumerated by the quotient Γ ∨ /Γ , where Γ ∨ is the dual lattice. Thus, in the one-dimensional case, we have m irreducible representations, to be denoted by Lm,i , 0  i  m − 1, in what follows. They can be described in terms of Fock modules over the Heisenberg algebra H , Lm,i =

 n∈Z

H nm+i , √ m

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where for any c ∈ C, Hc denotes the Fock module with the highest-weight vector |c. Note ± (z) act from H(nm+i)/√m to H(nm+i±m)/√m . Another that the Fourier coefficients of φ√ m important property is   √ ± ± (z) = ± m zk φ√ (z), hk , φ√ m m where hk , k ∈ Z, are the generators of the Heisenberg algebra. 3. Vertex operators and Jack polynomials We recall the definition of Jack polynomials (see [11]). Let Λ be the algebra of symmetric polynomials in infinitely many variables and pk ∈ Λ be the power sums, pk = i∈N xik . We define the scalar product on Λ = C[p1 , p2 , . . .] depending on a coupling constant α, 

p1i1 · · · pkik , p1i1 · · · pkik



= α− α

k

s=1 is

k 

s is is ! .

s=1

The products of power sums form an orthogonal basis of Λ with respect to ·,·α . Now let λ = (λ1  λ2  · · ·  λs  0) be a partition. Let l(λ) be the length of λ, i.e., the number of nonzero λi . The Young diagram attached to λ is the following subset of Z2 : {(i, j ): 1  i  l(λ), 1  j  λi }. Jack polynomials Jλ (α; x) = Jλ (α; x1 , x2 , . . .) ∈ Λ depend on the partition λ and the coupling constant α. They are uniquely determined by the properties Jλ (α, x) = mλ +



vλ,μ (α)mμ ,

μ<λ

  Jλ (α, x), Jμ (α, x) α = 0 if λ = μ, where mλ is the symmetrization of the monomial x1λ1 · · · xsλs and for two partitions λ and μ, we write μ  λ if μ1 + · · · + μi  λ1 + · · · + λi for any i. We note that Jack polynomials in a finite number of variables can be defined similarly. In that case, Jλ (α; x1 , . . . , xN ) = 0 if N < l(λ). We now introduce notation for the one-dimensional lattice VOA with even m. We set + (z), and L(m) , i = L2m,i . We use the notation an for the A(m) = A2m , φ√2m (z) = φ√ 2m Fourier coefficients of φ√2m (z): φ√2m (z) =



an z−n−m .

n∈Z

Let vp , p ∈ Z, be the set of extremal vectors of L(m),i :   2m−1   −p  ∈ vp =  √ L(m),i . 2m i=0

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To recall the connection between vertex operators and Jack polynomials, we first identify any Fock module Hc with Λ as  I : h−i1 · · · h−ik |c →

m 2

k pi 1 · · · pi k .

√ (Recall that Hc = C[h−1 , h−2 , . . .] · |c.) Then hi with i > 0 acts on Λ via i 2/m ∂/∂pi . Let Jλ (m; x) denote the vector I −1 (Jλ (m, x)). The following theorem was proved in [12]: Theorem 3.1. For any p > 0, the equality a0k vp = C · J((p−m(k−1)−1)k ) (m; x), where C is some nonzero constant and (N k ) is the partition (N, . . . , N ), holds in   H−p/√2m +k √2m . k Remark 3.1. Recall that each L(m),i is a representation of the Virasoro algebra, Ln =

1  m−1 : hj hl : − √ (n + 1)hn , 2 2m

(8)

j +l=n

√ where : : is the normal ordering sign. The factor (m − 1)/ 2m is fixed by the condition [a0 , Ln ] = 0. Note that in this case, the vectors a0k vp are singular vectors for the Virasoro algebra for any k  0. We introduce the characters of h0 - and L0 -invariant subspaces of L(m),i . Namely, for V → L(m),i we set   ch(z, q, V ) = Tr zh0 q L0 |V . For an eigenvector v of the operators h0 and L0 , let degz v and degq v denote the corresponding eigenvalues. For example, −p degz vp = √ , 2m

degq vp =

p(m − 1) p2 + . 4m 2m

We introduce the principal subspace V(m),p = C[ap−1 , ap−2 , . . .] · vp (note that ai vp = 0 for i  p because the difference of q-degrees of vp and vp−2m is equal to p − 1). We first study the space V(m) = V(m),−m+1 . + Lemma 3.1. Let φ√

2m

(z) = a−m + za−m−1 + z2 a−m−2 + · · · . Then

 + V(m)  C[a−m , a−m−1 , . . .] φ√

2m

(z)(i)

2  0im−1

,

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+ where φ√

2m

(z)(i) is the ith derivative. Therefore, the ideal of relations in V(m) is generated

+ by the coefficients of the series (φ√



2m

(z)(i) )2 , namely, by

a−m−α1 a−m−α2 α1 (α1 − 1) · · · (α1 − i + 1)α2 (α2 − 1) · · · (α2 − i + 1),

α1 +α2 =s α1 ,α2 0

s  0,

i = 0, . . . , m − 1.

Proof. Recall that φ√2m (z)φ√2m (w) = (z − w)2m : φ√2m (z)φ√2m (w) : . Therefore, because ai v1−m = 0 for i > −m, we obtain  + φ√

2m

2 (z)(i) v1−m = 0,

0  i  m − 1.

Because of Lemma 1.1, it suffices to prove that the character of V(m) is greater than or equal to (in each weight component)

z

m−1 √ 2m

q

−(m−1)2 4m

∞  k=0

2

zk q mk . (1 − q)(1 − q 2 ) · · · (1 − q k )

√ (Note that degq v1−m = −(m − 1)2 /(4m), degz v1−m = (m − 1)/ 2m .)  k,s , where Consider the decomposition V(m) = k,s0 V˜(m)  k,s V˜(m)

= span ai1 · · · aik v1−m ,

k 

 iα = −s .

α=1 k =  V˜ k,s . We identify (V˜ k )∗ with the subspace of symmetric polynomials in Let V˜(m) s (m) (m) k variables as



 k ∗ V˜(m)  θ → fθ (z1 , . . . , zk ) =

z1i1 · · · zkik θ (a−i1 · · · a−ik v1−m ).

i1 ,...,ik 0 + Because of the relations satisfied by φ√

2m

fθ (z1 , . . . , zk ) =

k  i=1

zim

(z), we obtain that fθ is of the form 

1i
(zi − zj )2m g(z1 , . . . , zk ),

(9)

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where g is some symmetric polynomial. Note that the character of the right-hand side of (9) with g ranging over all symmetric polynomials in k variables coincides with 2

q mk . (1 − q)(1 − q 2 ) · · · (1 − q k ) Therefore, we only need to prove that any symmetric g appears in the right-hand side of (9) for some θ . k,mk 2 We first note that V˜(m) is a subspace of the one-dimensional space spanned by the 2

k,mk is nontrivial. Therefore, vector v1−m−2k . At the same time, it is easy to show that V˜(m) 2

k,mk ∗ ) , for some nontrivial element θ ∈ (V˜(m)

fθ (z1 , . . . , zk ) =

k  i=1



zim

(zi − zj )2m .

1i
√ k . Let h∗ 2m ai+j , the operators hi with i > 0 act on V˜(m) i be the dual operators, acting on (V˜ k )∗ . Then Due to the relations [hi , aj ] =

(m)

fh∗i θ (z1 , . . . , zk ) =

√  i  2m z1 + · · · + zki fθ (z1 , . . . , zk ).

To finish the proof, it suffices to note that power sums generate the algebra of symmetric polynomials. 2 The following lemma can be easily verified. Lemma 3.2. The map C[a−m , a−m−1 , . . .] · v1−m → C[ap−1 , ap−2 , . . .] · vp , ai1 · · · aik v1−m → ai1 +m+p−1 · · · aik +m+p−1 vp is a well-defined isomorphism. Now we study the finitization of V(m),p . For p > 0, we define the subspace V(m),p ← V(m) (p) = C[ap−1 , ap−2 , . . . , a0 ] · vp , and introduce the notation   k,s V(m) (p) = v ∈ V(m) (p): h0 v = kv, L0 v = sv ,

k V(m) (p) =

 s

The following lemma is an immediate consequence of Lemma 3.2.

k,s V(m) (p).

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Lemma 3.3. Let V(m),p (n) = C[ap−1 , . . . , ap−n ] · vp . Then the map V(m),p (n) → V(m) (n),

ai1 · · · aik vp → ai1 −p+n · · · aik −p+n vn ,

is an isomorphism. √ √ −p/ 2m +k 2m

Lemma 3.4. V(m)

Proof. Note that [hi , aj ] =

(p) = C[h1 , h2 , . . .] · a0k vp .

√ 2m ai+j . Therefore,

C[h1 , h2 , . . .] · a0k vp

√−p 2m

→ V(m)

√ +k 2m

(p).

To show that the spaces are actually equal, we prove that for any p − 1  i1  i2  · · ·  ik  0, we have ai1 · · · aik vp ∈ C[h1 , h2 , . . .] · a0k vp . For this, we use the induction on r: the number of α such that iα = 0. The case r = 0 is trivial. Suppose the lemma is proved for some r. Then   √ hir+1 ai1 · · · air a0k−r vp = 2m (k − r)ai1 · · · air air+1 a0k−r−1 vp  r  √ + 2m a0k−r ai1 · · · aij +ir+1 · · · air vp . j =1

Therefore, by the induction hypothesis, ai1 · · · air air+1 a0k−r−1 vp ∈ C[h1 , h2 , . . .] · a0k vp . The lemma is proved.

2

As mentioned above, if we identify the Fock module with the space Λ of symmetric polynomials in infinitely many variables, then hi with i > 0 acts on Λ (up to a nonzero constant) as ∂/∂pi . Therefore, in view of the above lemma and Theorem 3.1, it is important to study the spaces  C

 ∂ ∂ , , . . . · Jλ (m; x) ∂p1 ∂p2

for rectangular diagrams λ. Consider a polynomial g ∈ C[x2 , x3 , . . .]. Let (Lg)(x1 , x2 , . . .) be the polynomial obtained from g(x2 , x3 , . . .) by replacing xi+1 with xi , i = 1, 2, . . . .

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Proposition 3.1. Let a collection of maps Tn : Λ → Λ, n = 1, 2, . . . , be defined by

 n    ∂ f  . Tn f (x1 , x2 , . . .) = L ∂x1n x1 =0 Then two algebras of operators on Λ are equal:  C

 ∂ ∂ , , . . . = C[T1 , T2 , . . .]. ∂p1 ∂p2

Proof. We prove that for any n > 0,  ∂ ∂ ∂ = C[T1 , T2 , . . . , Tn ]. C , ,..., ∂p1 ∂p2 ∂pn 

Let f ∈ Λ, f = f (p1 , p2 , . . .). Then there exist polynomials bj,n (k1 , . . . , kj ), j = 1, . . . , n, kj ∈ N, such that ∂ nf  = ∂x1n n



j =1 k1 ,...,kj 1 k1 +···+kj n

∂j f k +···+kj −n bj,n (k1 , . . . , kj )x1 1 , ∂pk1 · · · ∂pkj

b1,n (k1 ) = k1 (k1 − 1) · · · (k1 − n + 1).

(10)

We prove (10) by induction on n. For n = 1, we have  ∂f ∂pk  ∂f ∂f = = kx k−1 . ∂x1 ∂pk ∂x1 ∂pk 1 k1

k1

Now suppose (10) is true for n. Note that ∂ ∂x1



=

kn

∂f k(k − 1) · · · (k − n + 1)x1k−n ∂pk



 ∂f k(k − 1) · · · (k − n)x1k−n−1 ∂pk

kn+1

+

 k1 ,k2 1 k1 +k2 n+1

In addition,

∂ 2f k2 k1 (k1 − 1) · · · (k1 − n + 1)x1k1 +k2 −n−1 . ∂pk1 ∂pk2

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∂ ∂x1



 k1 +···+kj n k1 ,...,kj 1

∂j f k +···+kj −n bj,n (k1 , . . . , kj )x1 1 ∂pk1 · · · ∂kj



=

∂j f k +···+kj −n−1 bj,n (k1 , . . . , kj )(k1 + · · · + kj − n)x1 1 ∂pk1 · · · ∂pkj

k1 +···+kj n+1 k1 ,...,kj 1

+



 k1 +···+kj n k1 ,...,kj +1 1

∂ j +1 f k +···+kj +1 −n−1 kj +1 bj,n (k1 , . . . , kj )x1 1 . ∂pk1 · · · ∂pkj +1

Note that the sum in the last line ranges over k1 , . . . , kj +1 with k1 + · · · + kj +1  n + 1. Thus (10) is verified. Now consider L((∂ n f/∂x1n )|x1 =0 ). Because L(pi |xi =0 ) = pi , we obtain 

∂ n f  Tn (f ) = L ∂x1n x1 =0 

 ∂f n! + ∂pn n

=



j =2 k1 +···+kj =n

∂j f bj,n (k1 , . . . , kj ). ∂pk1 · · · ∂pkj

Now we can prove by induction on n that   ∂ ∂ = C[T1 , . . . , Tn ]. C ,..., ∂p1 ∂pn

(11)

(12)

For n = 1, we have  T1 f = L



∂f ∂f  = . ∂x1 x1 =0 ∂p1

In addition, if we know (12) for n, (11) yields the statement for n + 1. The proposition is proved. 2 Remark 3.2. For any f ∈ Λ, f (x1 , x2 , x3 , . . .) = f (x2 , x3 , . . .) +

∞  1 k x (Tk f )(x2 , x3 , . . .). k! 1 k=1

Recall the definition of skew Jack polynomials Jλ/μ (x). (From now on, we suppress the λ as coupling constant m in the notation for Jack and skew Jack polynomials.) Define fμν Jμ (x)Jν (x)Jμ , Jμ −1 Jν , Jν −1 =

 λ Jλ , Jλ −1 fμν Jλ (x), λ

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where λ, μ, and ν are some partitions. Then Jλ/μ (x) =



λ fμν Jν (x),

Jλ/μ (x1 ) =



ν

λ fμν Jν (x1 )

ν

(obviously, the summation in the second sum ranges over ν with l(ν)  1). For two partitions μ and λ, we write μ ⊂ λ if the Young diagram of μ is a subset of the Young diagram of λ. Note that Jλ/μ (x) = 0 unless μ ⊂ λ. The following proposition is proved in [11]. Proposition 3.2.  (a) Jλ (x1 , x2 , . . .) = μ Jλ/μ (x1 )Jμ (x2 , x3 , . . .). (b) Suppose that μ ⊂ λ. Then Jλ/μ (x1 ) = 0 unless there exists i such that λj = μj for j = i. In that case, λ −μi

Jλ/μ (x1 ) = φλμ x1 i

,

where φλμ is some nonzero constant. Remark 3.3. Note that φλμ depend on the coupling constant α and do not vanish for α being a natural number.  Corollary 3.1. Tk (Jλ (x)) = k! μ φλμ Jμ (x), where the sum ranges over partitions μ such that there exists i with λi − μi = k, λj = μj if j = i. Proposition 3.3. Let λ = (s r ) = (s, s, . . . , s ). Then C[ ∂p∂ 1 , ∂p∂ 2 , . . .] · Jλ (x) is spanned by   Jμ (x) with μ ⊂ λ. r Proof. In view of Corollary 3.1 and Proposition 3.1, we only need to prove that Jμ (x) → C[T1 , T2 , . . .] · Jλ (x) for any μ ⊂ λ. Let μ = (μ1  · · ·  μr ), μ1  s. We use the decreasing induction on j such that μj < s but μj −1 = s. In the case where j − 1 = r, we have μ = λ. Now let j  r. Let μ˜ be the partition with μ˜ j = s,

μ˜ i = μi for i = j.

Then by the induction assumption, Jμ˜ (x) ∈ C[T1 , T2 , . . .] · Jλ (x). Apply Ts−μj to Jμ˜ (x):  1 φμν Ts−μj Jμ˜ (x) = φμμ ˜ Jμ (x) + ˜ Jν (x), (s − μj )! ν where the sum in the right-hand side ranges over ν such that there exists i > j with μi − νi = s − μj ,

να = μ˜ α for α = i.

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But for any such ν, we have νj = s and hence Jν (x) ∈ C[T1 , T2 , . . .] · Jλ (x). This gives Jμ (x) ∈ C[T1 , T2 , . . .] · Jλ (x) (because φμμ ˜ = 0). The proposition is proved. 2 √ √ −p/ 2m +k 2m

We now return to the space V(m) √ √ −p/ 2m +k 2m

Theorem 3.2. V(m)

(p).

(p) has a basis of the form

Jμ (x),

  μ ⊂ (p − (k − 1)m − 1)k .

Proof. By Lemma 3.4, we have √ +k 2m

√−p 2m

V(m)

(p) = C[h1 , h2 , . . .] · a0k vp .

Because of the identification (up to a nonzero constant) of hi , i > 0, with ∂/∂pi , our theorem is a consequence of Theorem 3.1 and Proposition 3.3. 2 We finish this section with a remark on the fermionic vertex operators. Remark 3.4. Consider the vertex operator φ√2m−1 (z). Let bi be its Fourier components, bi bj = −bj bi . Let L be some irreducible representation of the corresponding vertex algebra A2m−1 and v ∈ L be an extremal vector. Take a number p! such that bp−1 v = 0 and bk v = 0 for any k  p.! We consider the principal subspace U = (bp−1 , bp−2 , . . .) · v and its finitization U (n) = (bp−1 , . . . , bp−n ) · v. The study of U can be provided by the same means as in the even case. On the other hand, one needs some other tools to understand (m) the structure of U (n). We have a conjecture concerning its dimension: dim U (n) = Fn . (Note that the case of A3 is considered in [13].)

4. Dimensions and characters In this section, we find the dimensions and characters of V(m) (p) and of the space of coinvariants.  For a partition μ, we set degq μ = q μi . Lemma 4.1. Let chs,r (q) =



μ⊂(s r ) degq

μ. Then chs,r (q) =

Proof. First note that chs,1 (q) = 1 + q + · · · + q s = chs,r (q) =

 sμ1 ···μr 0

q

r

i=1 μi

=

s+1 1

s 



j =0

μr =j sμ1 ···μr

q

q

s+r  r

q

.

. In addition,

r

i=1 μi

=

s  j =0

q rj chs−j,r−1 .

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To finish the proof, it is enough to note that 



s+r r

s 

= q

 q

j =0

jr

s +r −j −1 . r −1 q

2

Proposition 4.1. (a) ch(z, q, V(m) (p)) = zdegz vp (m)

(b) Let Fn



k0 z

√   k 2m q degq vp−2km p−(m−1)(k−1) . k q

be natural numbers defined by Fi(m) = i + 1,

(m) (m) Fi+m = Fi+m−1 + Fi(m) .

i = 0, 1, . . . , m − 1,

(m)

Then dim V(m),p (n) = Fn . Proof. Consider an element Jλ (x) ∈ H−p/√2m +k √2m . Note that  

λi Jλ (x). L0 Jλ (x) = degq vp−2km + i

Therefore, Theorem 3.2 and Lemma 4.1 imply assertion (a). Note that (b) follows from (a) because dim V(m) (p + m) =

 p + m − (m − 1)(k − 1)

k0

=



k0

+

k

p + m − 1 − (m − 1)(k − 1) k

 p + m − 1 − (m − 1)(k − 1)

k>0

k−1

= dim V(m) (p + m − 1) +

 p − (m − 1)(k − 1)

k0

k

= dim V(m) (p + m − 1) + dim V(m) (p). The proposition is proved.

2

Remark 4.1. There is an obvious embedding V(m),p (p − 1) → V(m) (p). Consider the quotient V(m) (p)/V(m),p (p − 1). A natural way to prove the formula for the dimension of V(m) (p) is to identify this quotient with the space V(m) (p − m). This was done in [4] for m = 1, but the straightforward generalization failed for m > 1.

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In the following proposition, we find the dimension and character of the space of coinvariants for V(m),m−1 . We denote V(m),m−1 simply by V(m) . Proposition 4.2. Define C(m) (n) = V(m) /span{ai V(m) , i  −m − n}. (2m)

(a) dim C(m) (n) = Fn . (b) Note that C(m) (n) inherits a bigrading from V(m) . Thus its character is well defined. We have 

 √   degz v1−m degq v1−m k 2m mk 2 n − (2m − 1)(k − 1) ch z, q, C(m) (n) = z q z q . (13) k q k0

Proof. (a) follows from the statement (see Corollary 1.1) that the images of the vectors a−i1 · · · a−ik v1−m ,

iα+1 − iα  2m, i1  m, ik  n + m − 1; k = 0, 1, . . .

form a basis of C(m) (n). To prove (b), note that it follows from Corollary 1.1 that for n  2m,     ch z, q, C(m) (n) = ch z, q, C(m) (n − 1) + zdegz v1−m q degq v1−m z

√

2m m+n−1

q

  ch z, q, C(m) (n − 2m) .

It is straightforward to check that this relation is satisfied by the right-hand side of (13).

2

5. Semi-infinite bases In this section, we construct monomial bases for the spaces L(m),i . Lemma 5.1. Recall that A(m) = C[ξ0 , ξ1 , . . .]/(ξ(z)(i) )2 0im−1 . Consider the subalgebra A1(m) → A(m) , A1(m) = ξ0 A(m) . Then the map ρ : A(m) → A1(m) ,

ξi1 · · · ξik → ξ0 ξi1 +2m · · · ξik +2m ,

is a well-defined isomorphism. Proof. Recall the embedding A(m) → F from Section 1. Note that ξ0 = ψ1 (0) · · · ψ2m (0). Thus we obtain that ξ0 ξi = 0 for i = 0, . . . , 2m − 1 and ρ is an isomorphism.

2

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Lemma 5.2. The map V(m),p → A(m) , aj1 · · · ajs vp → ξ−j1 +p−1 · · · ξ−js +p−1 , is a well-defined isomorphism. Proof. Follows from Lemmas 3.1 and 3.2.

2

Corollary 5.1. L(m),i has a basis labeled by the sequences (sj )j ∈Z , sj ∈ N ∪ {0}, with the following properties: (1) There exists n ∈ Z such that s−i+2mk−1 = 1 for k  n and sj = 0 if j  −i + 2mn − 1 and j + i + 1 is not divisible by 2m. (2) There exists N ∈ Z such that sj = 0 for j < N . (3) For any j ∈ Z, sj + · · · + sj +2m−1  1. √ Proof. We rescale the vectors vp = |−p/ 2m  to satisfy the equations ap−1 vp = vp−2m . To each sequence (sj )j ∈Z satisfying properties (1)–(3), we assign the vector (n comes from property (1))

  sj aj v−i+2nm ∈ L(m),i . (14) j <−i+2mn

Then because of (1), for any n1 > n,

   s ajj v−i+2n1 m =



j <−i+2n1 m

j <−i+2nm

s ajj v−i+2nm .

Note that L(m),i is the limit lim V(m),−i+2nm , n→∞

L(m),i = · · · → V(m),−i → V(m),−i+2m → V(m),−i+4m → · · ·

(15)

and each V(m),i is isomorphic to A(m) in the sense of Lemma 5.2. In addition, embeddings (15) are compatible with the isomorphism in Lemma 5.1. Thus Corollary 1.1 implies that vectors (14) form a basis of L(m),i . 2

6. Fusion procedure Let A(m) (n) → A(m) be the subalgebra generated by ξ0 , . . . , ξn−1 . Then A(m) (n)  C[ξ0 , . . . , ξn−1 ]/I(m) (n), where I(m) (n) is an ideal in C[ξ0 , . . . , ξn−1 ]. Now let Z = (z1 , . . . , zn ) ∈ Cn , zi = zj for i = j . Let I(m),Z (n) → C[ξ0 , . . . , ξn−1 ] be the ideal generated by the elements

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   ξ0 + ξ1 zi + · · · + ξn−1 zin−1 ξ0 + ξ1 zj + · · · + ξn−1 zjn−1 ,

1  i  j  n, |i − j | < m.

It was proved in [4] that if Z(ε) = (z1 (ε), . . . , zn (ε)) ∈ Cn is a family of points of Cn with the properties zi (ε) = zj (ε) (i = j ) and limε→0 zi (ε) = 0 (1  i  n), then limε→0 I(1),Z(ε) (n) = I(1) (n). The following conjecture generalizes this limit (fusion) procedure to the case of general m. Conjecture 6.1. Let Z(ε) = (z1 (ε), . . . , zn (ε)) ∈ Cn be a family with the following properties: (1) zi (ε) = zj (ε) (2)

lim

ε→0

for i = j

z2 (ε) − z1 (ε) = 0, z1 (ε)

lim zi (ε) = 0,

and

ε→0

lim

zi+1 (ε) − zi (ε)

ε→0 zi (ε) − zi−1 (ε)

1  i  n. = 0,

i = 2, . . . , n − 1.

Then limε→0 I(m),Z(ε) (n) = I(m) (n). We now consider the following version of the above construction. Let R(m) (n) be the ring C[y1 , . . . , yn ]/yi yj |i−j |
n 

zji yj ,

i = 0, . . . , n − 1.

j =1

Then for some ideal I˜(m),Z (n) → C[a0 , . . . , an−1 ], we have R(m) (n)  C[a0 , . . . , an−1 ]/I˜(m),Z (n). Let I˜(m) (n) denote the limit limε→0 I˜(m),Z(ε) (n), where the family Z(ε) satisfies the properties in Conjecture 6.1. Lemma 6.1. The homomorphism of algebras C[ξ0 , . . . , ξn−1 ]/I(m) (n) → C[a0 , . . . , an−1 ]/I˜(m) (n)

(16)

sending ξi to an−1−i is an isomorphism. Let W(m) (n) denote the right-hand side of (16). The above lemma and Conjecture 6.1 allow us to identify W(m) (n) with V(m) (n). Thus we obtain that L(m),i can be constructed as the limit of the finite-dimensional spaces L(m),i = W(m) (2m − i) → W(m) (4m − i) → · · · , similarly to the case m = 1 (see [5]).

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Thus, starting from finite-dimensional spaces obtained by the fusion procedure, we construct the infinite-dimensional space as an inductive limit and endow this limit with the structure of a representation of the lattice vertex algebra. We also note that Jack polynomials are encoded in the structure√ of W(m) (n) as a module √ −p/ 2m +k 2m

(p) be a basis over the annihilating Heisenberg operators. In fact, let v ∈ V(m) vector of the form Jμ (m; x) for some μ, d = degq v − degq vp−2mk . Fix some i1 , . . . , ir > 0 with i1 + · · · + ir = d. Then hi1 · · · hik v = c(i1 , . . . , ik )vp−2mk .

(17)

Thus (17) gives " m #k/2 c(i , . . . , i ) ∂ ∂ 1 k ··· Jμ (m; x) = . ∂pi1 ∂pik 2 i1 · · · ik Therefore, Jμ (m; x) is completely determined by the numbers c(i1 , . . . , ik ).

Acknowledgments E.F. thanks V. Dotsenko for helpful information about Jack polynomials. The first author was partially supported by the RFBR grants 02-01-01015 and 04-0100303, SS 2044.2003.2 and the INTAS grant 03-51-3350. The second named author was partially supported by the RFBR grant 03-01-00167.

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