On irreducible characters of the Iwahori-Hecke algebra in type $A$

TL;DR

This paper develops new formulas for irreducible characters of the Iwahori-Hecke algebra of type A using vertex operators, including explicit cases and duality-based formulas, with applications to super-characters and trace formulas.

Contribution

It introduces general formulas for irreducible characters of the Iwahori-Hecke algebra of type A, including explicit formulas for hooks and two-row partitions, and a determinant-based Murnaghan-Nakayama formula.

Findings

Derived explicit formulas for hook and two-row partitions.

Established a determinant type Murnaghan-Nakayama formula.

Provided new proofs for existing character formulas.

Abstract

We use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type $A$. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. Using duality, we also formulate a determinant type Murnaghan-Nakayama formula and give another proof of Ram's combinatorial Murnaghan-Nakayama formula. As applications, we study super-characters of the Iwahori-Hecke algebra as well as the bitrace of the regular representation and provide a simple proof of the Halverson-Luduc-Ram formula.