Theoretical Statistics and Mathematics Unit, ISI Delhi

August 21, 2019,
3:30 PM at Seminar Room 2

Speaker:
Samrith Ram,
IIIT Delhi

Title:
Singer Cycles and Unimodular Polynomial Matrices over Finite Fields

Abstract of Talk

Let $\Fq$ denote the finite field with $q$ elements where $q$ is a prime power. Denote by $\mathrm{GL}_n(\Fq)}$ the general linear group of $n\times n$ nonsingular matrices over $\Fq$. An element $A\in \mathrm{GL}_n(\Fq)$ is called a Singer cycle if its order is the maximum possible in $\mathrm{GL}_n(\Fq)$. Determining the number of block companion Singer cycles is a problem closely related to many other problems in enumerative combinatorics. We will outline some of them including connections with a theorem of Philip Hall on conjugacy class size in the general linear group and some results of Wilf and others on the probability of coprime polynomials over finite fields. We also discuss an unsolved enumeration problem on polynomial matrices that generalizes the problem of counting Singer cycles.

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