Dual Murnaghan-Nakayama rule for Hecke algebras in Type $A$

TL;DR

The paper addresses computing irreducible characters $\chi^{\lambda}_{\mu}(q)$ of the Iwahori-Hecke algebra $H_n(q)$ of type $A$ via a dual Murnaghan-Nakayama rule that reduces the upper partition $\lambda$ instead of the lower partition. It develops a combinatorial interpretation of the coefficient $C_{m,\rho}$ that appears in the expansion of $e_m(x)$ into generalized Hall-Littlewood functions $q_{\rho}(x;t)$, using brick tabloids and the vertex-operator framework for Schur functions. The main contribution is a refined, explicit recursion for $\chi^{\lambda}_{\mu}(q)$ (extending earlier work JL1) and a worked example showing the method yields concrete polynomials, e.g., $\chi^{(3,2,1)}_{(4,2)}(q)=-q^3+2q^2-q$. The results enable efficient computation of irreducible characters and deepen connections among Hecke algebras, symmetric functions, and vertex-operator techniques.

Abstract

Let $χ^λ_μ$ be the value of the irreducible character $χ^λ$ of the Hecke algebra of the symmetric group on the conjugacy class of type $μ$. The usual Murnaghan-Nakayama rule provides an iterative algorithm based on reduction of the lower partition $μ$. In this paper, we establish a dual Murnaghan-Nakayama rule for Hecke algebras of type $A$ using vertex operators by applying reduction to the upper partition $λ$. We formulate an explicit recursion of the dual Murnaghan-Nakayama rule by employing the combinatorial model of ``brick tabloids", which refines a previous result by two of us (J. Algebra 598 (2022), 24--47).

Figures (2)

• Figure 1: A brick tabloid of $10$ with $b_1=5$, $b_2=2$ and $b_3=3$.
• Figure :

Theorems & Definitions (11)

• Proposition 2.1
• Proposition 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Lemma 3.1
• proof
• Remark 3.2
• Theorem 3.3
• proof