Mirabolic Hecke algebras, Schur-Weyl duality and Frobenius character formulas

TL;DR

This paper introduces new algebraic structures and bases for mirabolic Hecke algebras, establishes dualities with quantum groups, and derives character formulas and recursive rules for their irreducible representations.

Contribution

It presents a new presentation and basis for mirabolic Hecke algebras, constructs their character tables, and establishes Schur--Weyl duality with quantum groups, along with Frobenius formulas.

Findings

New basis for mirabolic Hecke algebra introduced

Schur--Weyl duality with quantum group established

Frobenius character formulas derived

Abstract

We first introduce a new presentation for the mirabolic Hecke algebra $\mathscr{H}_{n,R}(q)$ over an arbitrary commutative ring $R$ and derive a new basis. Based on this presentation, specializing to the case of $\mathscr{H}_n(q)$ over the field $\mathbb{C}(q)$, we construct a basis for the cocenter of $\mathscr{H}_n(q)$, which facilitates the definition of its character table. We further establish a Schur--Weyl duality between $\mathscr{H}_n(q)$ and the quantum group $U_q(\mathfrak{gl}_r)$. As an application, we obtain Frobenius character formulas for the irreducible characters of $\mathscr{H}_n(q)$ within the ring of symmetric functions. Finally, we derive a recursive Murnaghan--Nakayama rule for the computation of the character table.