On generalized Kostka polynomials and the quantum Verlinde rule

Abstract

Here we propose a way to construct generalized Kostka polynomials. Namely, we construct an equivariant filtration on tensor products of irreducible representations. Further, we discuss properties of the filtration and the adjoint graded space. Finally, we apply the construction to computation of coinvariants of current algebras. * Partially supported by CRDF grant RM1-265 1 (0, 0, . . . , 0). It is natural to conjecture that actually dim π/A 2k−1 π = dim π/A 2k−1 (t 1 , . . . , t k )π.Still it is an open problem. In [5] it was proved for (2, 2s + 1)-minimal theories, and in [6] for k = 2. Suppose that all the points t 1 , . . . , t k are distinct. In this case the dimension of π/A 2k−1 (t 1 , . . . , t k )π is given by the Verlinde formula (see [1]). More precisely, let V N be the Verlinde algebra, {π 1 , . . . , π N } be the standard basis in V N corresponding to the set of "integrable" representations. Let us write down in V N

Demazure flags, q-Fibonacci polynomials and hypergeometric series

“…6 We now state the more general version of Theorem 1; the proof of this theorem can be found in the next section. Remark (1) More generally, the isomorphism in Theorem 3 is of Cg-modules if we replace the tensor product by the fusion product; we refer the reader to [9] for the definition and properties of fusion products. Hence our result gives a presentation of a certain class of fusion products of different level Demazure modules.…”

The Serpentine Representation of the Infinite Symmetric Group and the Basic Representation of the Affine Lie Algebra $${\widehat{\mathfrak{sl}_2}}$$ sl 2 ^

“…Fusion product. Our proof relies on the result of B. Feigin and E. Feigin [2] on a finite-dimensional approximation of the basic representation of sl 2 , which, in turn, uses the notion of the fusion product of representations introduced in [4]. Since the corresponding construction is of importance for us, we describe it in some detail.…”

Proof of the Combinatorial Kirillov-Reshetikhin Conjecture

“…As g-modules, they are known to have the same structure as the Yangian KR-modules in the case of the classical Lie algebras, and in certain exceptional cases. These modules arise naturally when one considers the explicit description of the dual space of functions [1] to Feigin-Loktev fusion product [8].…”

“…The Feigin-Loktev conjecture and Kirillov-Reshetikhin conjecture. In their paper [8] the authors introduced a graded tensor product on finite-dimensional, graded, cyclic g[t]-modules for g a simple Lie algebra, which they call the fusion product. This is related to the fusion product in WessZumino-Witten conformal field theory when the level is restricted.…”

“…Let us summarize the results of [1] concerning the Feigin-Loktev conjecture for unrestricted fusion products of Kirillov-Reshetikhin modules for any simple Lie algebra g. The description below of the Feigin-Loktev fusion product [8] is the one given in [15,1]. We refer the reader to those articles for further details.…”