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Generating functions for the powers in $\text{GL}(n,q)$

Rijubrata Kundu, Anupam Singh

Abstract

Consider the set of all powers $\text{GL}(n ,q)^M = \{x^M \mid x\in \text{GL}(n, q)\}$ for an integer $M\geq 2$. In this article, we aim to enumerate the regular, regular semisimple and semisimple elements as well as conjugacy classes in the set $\text{GL}(n, q)^M$, i.e., the elements or classes of these kinds which are $M^{th}$ powers. We get the generating functions for (i) regular and regular semisimple elements (and classes) when $(q,M)=1$, (ii) for semisimple elements and all elements (and classes) when $M$ is a prime power and $(q,M)=1$, and (iii) for all kinds when $M$ is a prime and $q$ is a power of $M$.

Generating functions for the powers in $\text{GL}(n,q)$

Abstract

Consider the set of all powers for an integer . In this article, we aim to enumerate the regular, regular semisimple and semisimple elements as well as conjugacy classes in the set , i.e., the elements or classes of these kinds which are powers. We get the generating functions for (i) regular and regular semisimple elements (and classes) when , (ii) for semisimple elements and all elements (and classes) when is a prime power and , and (iii) for all kinds when is a prime and is a power of .

Paper Structure

This paper contains 20 sections, 45 theorems, 100 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Organisation of the paper
  4. Cycle index in $\text{GL}(n, q)$
  5. Regular, Semisimple and Regular Semisimple elements
  6. The main problem
  7. $M$-power polynomials
  8. $(q,M)=1$ and $M$ is a prime power
  9. $(M,q)=1$ and $M$ a prime
  10. $M^{th}$ powers in $\text{GL}(n,q)$
  11. $M^{th}$ power regular semisimple and regular classes in $\text{GL}(n,q)$
  12. $M^{th}$ power semisimple classes in $\text{GL}(n,q)$ when $M$ is a prime power
  13. $M^{th}$ power conjugacy classes in $\text{GL}(n, q)$ when $M$ is a prime power
  14. Generating functions when $M$ is a prime and $(M,q)=1$
  15. Generating function for regular and regular semisimple conjugacy classes
  16. ...and 5 more sections

Key Result

Proposition 3.3

For $d> 1$ we have,

Theorems & Definitions (91)

  • Definition 3.1: M-power polynomial
  • Example 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4: Butler
  • Lemma 3.5
  • proof
  • Corollary 3.6
  • proof
  • Lemma 3.7
  • ...and 81 more