Principle subspace for bosonic vertex operator $phi_{sqrt{2m}}(z)$ and Jack polynomials.pdf

a r X i v : m a t h / 0 4 0 7 3 7 2 v 1 [ m a t h .Q A ] 2 2 J u l 2 0 0 4 PRINCIPLE SUBSPACE FOR BOSONIC VERTEX OPERATOR φ√2m(z) AND JACK POLYNOMIALS B.FEIGIN AND E.FEIGIN Abstract. Let φ√2m (z) = ∑ n∈Z anz ?n?m, m ∈ N be bosonic vertex oper- ator, L some irreducible representation of the vertex algebra A(m), associated with one-dimensional lattice Zl, generated by the vector l, 〈l, l〉 = 2m. Fix some extremal vector v ∈ L. We study the principal subspace C[ai]i∈Z · v and its finitization C[ai]iN · v. We construct their bases and find characters. In the case of finitization basis is given in terms of Jack polynomials. Introduction Let L0,1 and L1,1 be irreducible representations of the Lie algebra s?l2, v2n ∈ L0,1, v2n+1 ∈ L1,1 the set of extremal vectors (for example, v0 is a vacuum vector). For x ∈ sl2 consider the current x(z) = ∑ i∈Z xiz ?i?1 (here we use the notation xi = x? ti). Let e, h, f be standard basis of sl2. Then eivp = 0 for i ≥ p. Consider the principle subspace Vp = C[ep?1, ep?2, . . .] · vp. Let us list some properties of Vp (see [FL, FF1, FF2, CP]). 1) Vp ? C[ep?1, ep?2, . . .]/Ip, where Ip is an ideal, generated by coefficients of series (ep?1 + zep?2 + z2ep?3 + . . .)2. 2) Elements ei1 · · · eikv, i1 p, iα ? iα+1 ≥ 2, k = 0, 1, . . . form the basis of Vp. Using this basis one can write formula for the character of Vp and construct semi-infinite basis of L0,1 and L1,1. 3) Consider finitization: the subspace Vp(n) →? Vp, Vp(n) = C[ep?1, . . . , ep?n] · vp. Then dimVp(n) = 2 n. Recall that the current e(z) can be realized as bosonic vertex operator φ√2(z). In this paper we generalize the above results to the case of φ√2m(z) for an arbitrary m ∈ N (see [FJM] for the discussion on this topic). Let A(m) be lattice vertex algebra, associated with a one-dimensional lattice Zl, generated by vector l , 〈l, l〉 = 2m. Let L(m),i be the set of irreducible representa- tions of A(m) (see [D]). We have L(m),i = ⊕ n∈Z H 2nm+i√ 2m , 0 ≤ i ≤ 2m? 1. Here Hλ