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Irreducible characters and bitrace for the $q$-rook monoid

Naihuan Jing, Yu Wu, Ning Liu

TL;DR

The paper develops a computation-friendly character theory for the $q$-rook monoid algebra $R_n(q)$ by employing vertex operator techniques to produce a new iterative formula for irreducible characters $\\chi^{\lambda}_{\mu}(q)$ and a rederivation of the Murnaghan–Nakayama rule. It introduces compact closed forms for hook and two-row shapes and extends the framework with the bitrace, a deformed orthogonality relation, providing a general combinatorial formula based on weighted integer matrices. The approach hinges on a Frobenius-type formula connecting characters to modified Hall–Littlewood/ribbon-type symmetric functions and the vertex operator realization of Schur functions. The work yields practical computational tools and includes explicit character data for $|\mu|=5$ in the appendix, broadening the understanding of $R_n(q)$-representations and their connections to Hecke-algebraic structures.

Abstract

This paper studies irreducible characters of the $q$-rook monoid algebra $R_n(q)$ using the vertex algebraic method. Based on the Frobenius formula for $R_n(q)$, a new iterative character formula is derived with the help of the vertex operator realization of the Schur symmetric function. The same idea also leads to a simple proof of the Murnaghan-Nakayama rule for $R_n(q)$. We also introduce the bitrace for the $q$-rook monoid and derive its combinatorial formula as a generalization of the bitrace formula for the Iwahori-Hecke algebra. The character table of $R_n(q)$ with $|μ|=5$ is listed in the appendix.

Irreducible characters and bitrace for the $q$-rook monoid

TL;DR

The paper develops a computation-friendly character theory for the -rook monoid algebra by employing vertex operator techniques to produce a new iterative formula for irreducible characters and a rederivation of the Murnaghan–Nakayama rule. It introduces compact closed forms for hook and two-row shapes and extends the framework with the bitrace, a deformed orthogonality relation, providing a general combinatorial formula based on weighted integer matrices. The approach hinges on a Frobenius-type formula connecting characters to modified Hall–Littlewood/ribbon-type symmetric functions and the vertex operator realization of Schur functions. The work yields practical computational tools and includes explicit character data for in the appendix, broadening the understanding of -representations and their connections to Hecke-algebraic structures.

Abstract

This paper studies irreducible characters of the -rook monoid algebra using the vertex algebraic method. Based on the Frobenius formula for , a new iterative character formula is derived with the help of the vertex operator realization of the Schur symmetric function. The same idea also leads to a simple proof of the Murnaghan-Nakayama rule for . We also introduce the bitrace for the -rook monoid and derive its combinatorial formula as a generalization of the bitrace formula for the Iwahori-Hecke algebra. The character table of with is listed in the appendix.
Paper Structure (11 sections, 13 theorems, 82 equations, 1 table)

This paper contains 11 sections, 13 theorems, 82 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partitions and symmetric functions
  4. Vertex operator realization of Schur functions
  5. Irreducible characters for the $q$-rook monoid algebra
  6. Irreducible characters
  7. A new iterative formula
  8. Murnaghan-Nakayama rule
  9. Compact formulae for hook and two-row types
  10. The second orthogonal relation
  11. A table of $\chi^{\lambda}_{\mu}(q)$

Key Result

Proposition 2.1

Jing1Jing2 (1) The components of $S(z)$ and $S^*(z)$ obey the following commutation relations: (2) For any composition $\mu=(\mu_{1},\ldots,\mu_{k})$, the product $S_{\mu_{1}}\cdots S_{\mu_{k}}.1=s_{\mu}$ is the Schur function labeled by $\mu$. In general, $s_{\mu}=0$ or $\pm s_{\lambda}$ for a partition $\lambda$ such that $\lambda=\sigma(\mu+\delta)-\delta$ for some $\sigma\in\mathfrak S_k$

Theorems & Definitions (26)

  • Proposition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Example 3.4
  • ...and 16 more