Character factorizations for representations of GL(n,C)

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 ${\omega }_n^{j-1}{t}_i$ for $1\le i\le m$, $1\le j\le n$, where ${\omega }_n=e^{2\pi ı/n}$, expressing it as a product of certain characters for $\GL(m,\C)$ evaluated at ${\underline{t}}^n=\mathrm{diag}(t_1^n,t_2^n,\dots ,t_m^n)$. Unlike previous approaches that rely on determinantal identities, our proof utilizes a direct combinatorial cancellation argument within the Weyl group.