Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jinkui Wan

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.

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

Abstract

We first introduce a new presentation for the mirabolic Hecke algebra over an arbitrary commutative ring and derive a new basis. Based on this presentation, specializing to the case of over the field , we construct a basis for the cocenter of , which facilitates the definition of its character table. We further establish a Schur--Weyl duality between and the quantum group . As an application, we obtain Frobenius character formulas for the irreducible characters of within the ring of symmetric functions. Finally, we derive a recursive Murnaghan--Nakayama rule for the computation of the character table.
Paper Structure (8 sections, 23 theorems, 95 equations)

This paper contains 8 sections, 23 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. The mirabolic Hecke algebra $\mathscr{H}_{n,R}(q)$
  3. The mirabolic Hecke algebra $\mathscr{H}_{n,R}(q)$.
  4. A new presentation for $\mathscr{H}_{n,R}(q)$.
  5. The space of trace functions and character table of $\mathscr{H}_n(q)$.
  6. Schur-Weyl duality between $U_q(\mathfrak{gl}_r)$ and $\mathscr{H}_n(q)$
  7. Frobenius character formulas for $\mathscr{H}_n(q)$
  8. Murnaghan-Nakayama rule for characters of $\mathscr{H}_n(q)$

Key Result

Lemma 2.2

SiSo1 For each $\omega\in\Lambda_n$, the element $T_\omega$ only depends on $\omega$. The set $\{T_{\omega}\mid \omega\in\Lambda_n\}$ is a $\mathbb{C}(q)$-basis of $\mathscr{H}_n(q)$ and moreover

Theorems & Definitions (43)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • ...and 33 more