Spectral Separation and Eigenvalue Labelling for Polynomial Tensor Representations of General Linear Groups

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 $\bigotimes L(\lambda^{(t)})$ 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 $K<q-1$. These results provide a uniform spectral explanation for eigenvalue separation in bounded-degree polynomial tensor representations. Motivated by this, we formulate a conditional rewriting framework that uses compatible base-$q$ eigenvalue labelling to reduce the reconstruction of the natural action to a functor-specific inversion problem. Finally, the viability of this framework is explicitly demonstrated through computational experiments, prominently featuring a non-trivial, full algebraic reconstruction of the natural action from a strictly multiplicity-free, genuine tensor product representation.