Fermionic Formulas for Level-Restricted Generalized Kostka Polynomials and Coset Branching Functions

Abstract

Abstract. Level-restricted paths play an important rôle in crystal theory. They correspond to certain highest weight vectors of modules of quantum affine algebras. We show that the recently established bijection between Littlewood-Richardson tableaux and rigged configurations is well-behaved with respect to level-restriction and give an explicit characterization of level-restricted rigged configurations. As a consequence a new general fermionic formula for the level-restricted generalized Kostka polynomial is obtained. Some coset branching functions of type A are computed by taking limits of these fermionic formulas.

Paths, Crystals and Fermionic Formulae

“…Before going into the contents of the paper, we note further aspects of the subject including some latest ones; applications to quasi-hypergeometric functions [AI], exclusion statistics [BM,BS2], decomposition of the KirillovReshetikhin modules W (k) s (see Section 2.3) into classical irreducible modules [C1], the Feigin-Stoyanovsky theory [FS], fermionic formulae for the spaces of coinvariants associated to sl 2 [FKLMM], modular representations of Hecke algebras [FLOTW], approaches by various bijections and tableaux [FOW,KSS,SW,S,SS1], a graphical computational algorithm of fermionic formulae [Kl], dilogarithm identities [Ki4], fermionic formulae for weight multiplicities [KN], connection to geometry of quiver varieties [Lu, N], spinons at q = 0 [NY3,NY4], fermionic formulae from fixed boundary RSOS models [OPW], decomposition of level 1 modules by level 0 actions [Ta,KKN,BS1], bosonic formulae from crystals [SS2,KMOTU], etc.…”

Fusion products, Kostka polynomials and fermionic characters of

“…2 Level-restricted generalized Kostka polynomials [18] are also important, but do not play a role in our calculations.…”

K-matrices for 2D conformal field theories

“…Obviously, the constraints on which fusion rules correspond to which pseudoparticle K-matrix derived this way are much weaker than those arising from the comparison of the above recursion relations. Consider a CFT with two primary fields 1 and φ and nontrivial fusion rule φ × φ = 1, i.e., 42) which has eigenvalues λ = ±1 and is diagonalized by…”