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Counting generating spaces of matrices

Markus Reineke

TL;DR

The paper addresses counting generating subspaces of matrix algebras over finite fields and connects these counts to absolutely irreducible representations of free algebras. It introduces counting polynomials $s_d^{(m)}(q)$ for generating subspaces and links them to $a_d^{(m)}(q)$ via a detailed representation-theoretic and geometric framework, including Exp/Log methods and Gaussian-binomial identities. A two-variable extension $a_d(q,u)$ is constructed to capture rank-dependent counts, yielding a Mahler-type expansion in terms of the $s_d^{(r)}(q)$ and revealing structural factorization and term behavior. The results collectively establish polynomial-count phenomena and provide explicit formulas and recursions, enriching the understanding of polynomiality in finite-field counting problems and connecting to broader themes in quiver and representation theory.

Abstract

We prove that generating subspaces of matrix rings over finite fields are counted by polynomials. We use this result to define and study two-variable versions of polynomials counting isomorphism classes of absolutely irreducible representations of free algebras.

Counting generating spaces of matrices

TL;DR

The paper addresses counting generating subspaces of matrix algebras over finite fields and connects these counts to absolutely irreducible representations of free algebras. It introduces counting polynomials for generating subspaces and links them to via a detailed representation-theoretic and geometric framework, including Exp/Log methods and Gaussian-binomial identities. A two-variable extension is constructed to capture rank-dependent counts, yielding a Mahler-type expansion in terms of the and revealing structural factorization and term behavior. The results collectively establish polynomial-count phenomena and provide explicit formulas and recursions, enriching the understanding of polynomiality in finite-field counting problems and connecting to broader themes in quiver and representation theory.

Abstract

We prove that generating subspaces of matrix rings over finite fields are counted by polynomials. We use this result to define and study two-variable versions of polynomials counting isomorphism classes of absolutely irreducible representations of free algebras.

Paper Structure

This paper contains 4 sections, 16 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Polynomials counting generating subspaces
  4. Two-parameter versions of counting polynomials for absolutely irreducible representations

Key Result

Theorem 2.1

There exist polynomials $a_d^{(m)}(q)\in\mathbb{Z}[q]$ such that $a_d^{(m)}(|k|)$ equals the number of isomorphism classes of $d$-dimensional absolutely irreducible representations of $F^m(k)$ for all finite fields $k$.

Theorems & Definitions (30)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • ...and 20 more