Character factorizations for representations of GL(n,C)

We give another proof of a theorem of D. Prasad (Theorem 2, $\mathit{\text{Israel}}\mathit{\text{J.}}\mathit{\text{Math.}}$ 2016), which is also a classical result of Littlewood–Richardson (Theorem VI, $\mathit{\text{Q.}}\mathit{\text{J.}}\mathit{\text{Math.}}$ 1934). For integers $m,n\ge 2$, this result calculates the character of an irreducible representation of $\text{GL}(mn,\text{C})$ at diagonal elements with eigenvalues ${\omega }_n^{j-1}{t}_i$ for $1\le i\le m$, $1\le j\le n$, where ${\omega }_n=e^{2\pi \mathrm{ı}\mathrm{/}n}$, expressing it as a product of certain characters for $\text{GL}(m,\text{C})$ evaluated at ${\underset{‾}{t}}^n=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t_1^n,t_2^n,\dots ,t_m^n)$. Unlike previous approaches that rely on determinantal identities, our proof utilizes a direct combinatorial cancellation argument within the Weyl group.