Representation theory of mirabolic quantum $\mathfrak{sl}_n$

TL;DR

This work identifies the mirabolic quantum group $MU(n)$ as a comodule algebra over the quantized enveloping algebra $U_v(\mathfrak{sl}_n)$ and achieves a complete classification of its finite dimensional representations. It constructs irreducible modules $L_{\lambda,r}$ parameterized by a dominant weight $\lambda$ and an integer $0\le r\le n$, and proves the category of finite dimensional $MU(n)$-modules is semisimple. The paper also establishes a detailed mirabolic quantum Schur–Weyl duality between $MU(n)$ and the mirabolic Hecke algebra $MH_d$ via the bimodule structure of the mirabolic tensor space $MV_{n,d}$, giving an explicit correspondence between simple modules: $L_{\lambda,r}$ pairs with $M^{\boldsymbol\lambda}$ for bipartitions $\boldsymbol\lambda=(\lambda,1^r)$ in the set $M\Lambda_{n,d}$. The results extend quantum Schur–Weyl theory to the mirabolic setting and provide explicit action formulas, enabling a concrete combinatorial and representation-theoretic understanding with potential extensions to roots of unity and categorification.

Abstract

We show that the mirabolic quantum group $MU(n)$ is a comodule algebra over the quantized enveloping algebra $U_v(\mathfrak{sl}_n)$, and use this structure to give a complete classification of its finite dimensional representations. We also explicitly describe the correspondence between the irreducible finite dimensional representations of $MU(n)$ and the ones for the mirabolic Hecke algebra, given by the mirabolic quantum Schur-Weyl duality.

Theorems & Definitions (92)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Proposition 2.7
• proof
• Definition 2.8
• Definition 2.9