Symmetric Crystals and LLTA Type Conjectures for the Affine Hecke Algebras of Type B

TL;DR

This paper explores the connections between symmetric crystals and affine Hecke algebras of type B, proposing conjectures and reviewing related theorems for type A and B.

Contribution

It formulates new conjectures relating irreducible representations of affine Hecke algebras of type B to symmetric crystals, extending prior work on type A.

Findings

Survey of LLTA theorem for type A affine Hecke algebra

Construction of symmetric crystals for type B

Formulation of LLTA type conjectures for type B

Abstract

In the previous paper "Symmetric Crystals and Affine Hecke Algebras of Type B", we formulated a conjecture on the relations between certain classes of irreducible representations of affine Hecke algebras of type B and symmetric crystals for $\gl_\infty$. In the first half of this paper (sections 2 and 3), we give a survey of the LLTA type theorem of the affine Hecke algebra of type $A$. In the latter half (sections 4, 5 and 6), we review the construction of the symmetric crystals and the LLTA type conjectures for the affine Hecke algebra of type $B$.