Spectral Separation and Eigenvalue Labelling for Polynomial Tensor Representations of General Linear Groups
Year: 2025
Type: preprint
Abstract: Let $q=p^f$ be a prime power, $H \leq \mathrm{GL}_d(q)$ a subgroup containing a genuine Singer cycle $s$ of order $q^d-1$, and $W$ an $\mathbb{F}_q H$-module whose scalar extension restricts to an untwisted polynomial tensor representation $\bigotimes L(λ^{(t)})$ of the algebraic group $\mathrm{GL}_d$. If the total polynomial degree satisfies $K < q-1$, we prove that distinct weights give distinct eigenvalues of $s$ on $W \otimes_{\mathbb{F}_q} \mathbb{F}_{q^d}$. The proof relies on an elementary base-$q$ injectivity lemma: bounded digit vectors determine distinct residues modulo $q^d-1$. Consequently, when the tensor product is multiplicity-free for the diagonal torus, the Singer cycle has a simple spectrum. We also provide a shifted exponent formula for situations where Singer eigenvalue data undergo $q$-Frobenius shifts, proving separation of distinct shifted digit vectors under the same bound $K
Source: arXiv (Cornell University)
Author: Phuc, Dang Vo
Institutions: FPT University, Quy Nhon University
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416933283
Cites:
Cited by:
Related to: