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 $ω^{j-1}_nt_i$ for $1 \le i \le m$, $1 \le j \le n$, where $ω_n=e^{2π\imath/n}$, 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.