Character factorizations for representations of GL(n,C)

TL;DR

This paper provides a new combinatorial proof of a classical character formula for irreducible representations of GL(mn,C), relating it to characters of GL(m,C) evaluated at specific diagonal matrices.

Contribution

It introduces a direct combinatorial cancellation method within the Weyl group to prove a character formula, avoiding determinantal identities used in prior proofs.

Findings

Reproduces a known character formula for GL(mn,C) representations.

Uses combinatorial cancellation in the Weyl group instead of determinantal identities.

Offers a new proof technique for classical results in representation theory.

Abstract

We give another proof of a theorem of D. Prasad (Theorem 2, \textit{Israel J. Math.} 2016), which is also a classical result of Littlewood--Richardson (Theorem VI, \textit{Q. J. Math.} 1934). For integers $m,n \ge 2$, this result calculates the character of an irreducible representation of $\GL(mn,\C)$ at diagonal elements with eigenvalues $\omega^{j-1}_nt_i$ for $1 \le i \le m$, $1 \le j \le n$, where $\omega_n=e^{2\pi \imath/n}$, expressing it as a product of certain characters for $\GL(m,\C)$ evaluated at $\underline{t}^n={\rm diag}(t_1^{n},t_{2}^{n},\dots,t_{m}^{n})$. Unlike previous approaches that rely on determinantal identities, our proof utilizes a direct combinatorial cancellation argument within the Weyl group.