A Uniform Rewriting Algorithm for Twisted Tensor Representations of Finite General Linear Groups
Dang Vo Phuc
TL;DR
The work tackles the problem of rewriting irreducible F_qG-modules, arising as twisted polynomial tensor products with total degree K<q-1, to the natural d-dimensional representation of SL_d(q) or GL_d(q). It introduces a base-q injectivity lemma and a uniform spectral analysis of Singer-cycle actions, enabling a Las Vegas rewriting algorithm for modules multiplicity-free for the diagonal torus. The algorithm runs in polynomial time in d, log q, and K, and extends prior rewriting frameworks to a broad class of polynomial tensor modules without case-by-case eigenvalue analyses. This advances constructive recognition by providing a versatile, uniform tool that integrates with existing matrix-group recognition pipelines to realize isomorphisms to standard classical groups in diverse settings.
Abstract
Let $q=p^f$ be a prime power and let $H$ be a classical group of type $A$, so $H \cong \mathrm{SL}_d(q)$ or $\mathrm{GL}_d(q)$, acting on its natural module $V$ of dimension $d$ over $\mathbb{F}_q$. Let $W$ be an absolutely irreducible $\mathbb{F}_qH$--module such that, over an algebraic closure, $W$ is isomorphic (up to Frobenius twists) to a tensor product $\bigotimes_{t=1}^r L(λ^{(t)})$, where each $L(λ^{(t)})$ is an irreducible polynomial representation of $\mathrm{GL}_d$ of degree $k_t$, and the total degree $K=\sum_t k_t$ satisfies $K<q-1$. We prove a base-$q$ injectivity lemma which shows that, for a Singer cycle $s \in H$, the eigenvalues of $s$ on $W \otimes_{\mathbb{F}_q} \mathbb{F}_{q^d}$ are parametrised by digit vectors of length $d$ with entries bounded by $K$. In particular, distinct weights of $W$ correspond to distinct eigenvalues. If in addition $W$ is multiplicity-free for the diagonal torus of $\mathrm{GL}_d$, then every eigenspace of $s$ is one-dimensional, so $s$ has a simple spectrum on $W$. Using this spectral property we design a Las Vegas rewriting algorithm which, given $G \leq \mathrm{GL}(W)$ isomorphic to $H$ and acting irreducibly on $W$, constructs a projective representation of $G$ of degree $d$ equivalent to the natural representation of $H$. The expected running time is polynomial in $d$, $\log q$ and $K$, and in the cost of field operations and discrete logarithm computations in $\mathbb{F}_{q^d}$. This extends earlier rewriting algorithms of Magaard--O'Brien--Seress and of Gül--Ankaralıoğlu to a broader class of twisted tensor products of polynomial modules.