Dual canonical bases, quantum shuffles and q-characters

TL;DR

This paper investigates the properties of the dual canonical basis of the positive part of a quantum group when embedded into a quantum shuffle algebra, linking it to q-analogues of characters in type A_r.

Contribution

It analyzes the image of the dual canonical basis under the embedding, revealing connections to q-analogues of irreducible characters of affine Iwahori-Hecke algebras.

Findings

Properties of the embedded dual canonical basis are characterized.

Connections established between quantum bases and affine Iwahori-Hecke algebra characters.

Provides insights into the structure of quantum shuffle algebra representations.

Abstract

Rosso and Green have shown how to embed the positive part $U_q(n)$ of a quantum enveloping algebra $U_q(g)$ in a quantum shuffle algebra. In this paper we study some properties of the image of the dual canonical basis $B^*$ of $U_q(n)$ under this embedding $\Phi$. This is motivated by the fact that when $g$ is of type $A_r$, the elements of $\Phi(B^*)$ are $q$-analogues of irreducible characters of the affine Iwahori-Hecke algebras attached to the groups $GL(m)$ over a $p$-adic field.