Two Maps on Affine Type $A$ Crystals and Hecke Algebras

We use the crystal isomorphisms of the Fock space to describe two maps on partitions and multipartitions which naturally appear in the crystal basis theory for quantum groups in affine type $A$ and in the representation theory of Hecke algebras of type $G(l,l,n)$.

Global crystal bases and q-Schur algebras

We prove that the quantized Carter–Lusztig basis for a finite-dimensional irreducible Uq(𝔤𝔩n(ℂ))-module V(λ) is related to the global crystal basis for V(λ) by an upper triangular invertible matrix. We express the global crystal basis in terms of the q-Schur algebra and provide an algorithm for obtaining global crystal basis vectors for V(λ) using the q-Schur algebra.

Statement B and the Yang-Baxter Equation

This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.