Character formulae of sl̂n-modules and inhomogeneous paths

Abstract

Let B (l) be the perfect crystal for the l-symmetric tensor representation of the quantum affine algebra U ′ q ( sl n). For a partition µ = (µ1, . . . , µm), elements of the tensor product B (µ 1 ) ⊗· · ·⊗B (µm) can be regarded as inhomogeneous paths. We establish a bijection between a certain large µ limit of this crystal and the crystal of an (generally reducible) integrable Uq( sl n)-module, which forms a large family depending on the inhomogeneity of µ kept in the limit. For the associated one dimensional sums, relations with the Kostka-Foulkes polynomials are clarified, and new fermionic formulae are presented. By combining their limits with the bijection, we prove or conjecture several formulae for the string functions, branching functions, coset branching functions and spinon character formula of both vertex and RSOS types.

Paths, Crystals and Fermionic Formulae

“…In terms of appropriate summation variables 14 , Remark 4.7 reproduces the many expressions argued in [KKMM2,BLS2,NY2,HKKOTY] for simply-laced g. In the next two examples we let (C ij ) l−1 i,j=1 denote the Cartan matrix of A l−1 .…”

An Invitation to the Generalized Saturation Conjecture

“…See [32], Sections 3 and 4, and [22] , where further details and applications of the fermionic formula (5.57) can be found. In particular, the fermionic formula (5.57) gives an explicit expression for the number |F …”

Spinons in magnetic chains of arbitrary spins at finite temperatures

“…There are independent justifications of this: The double feature in the spectral decomposition is shown by path space approach [9] . See also [10] for the decomposition of space picture realized in Fermionic forms.…”