Chennai Mathematical Institute


2:00 pm, Lecture Hall 3
Splitting subspaces of linear transformations over finite fields

Samrith Ram
HRI, Allahabad.


Let m, n be positive integers and denote by Fq the finite field with q elements. Let V be a vector space of dimension mn over Fq and T : V -> V be a linear transformation. An m-dimensional subspace W of V is said to be T -splitting if

V = W (+) T W (+) · · · (+) T^{n-1} W.

Determining the number of m-dimensional T-splitting subspaces for an arbitrary transformation T is an open problem closely related to many problems in combinatorics and cryptography. I will outline connections with a theorem of Philip Hall on conjugacy class size in the general linear group and some results of Wilf et al. on the probability of coprime polynomials over finite fields. I will also discuss a general enumeration problem on matrix polynomials which, if solved, would settle the problem of counting T-splitting subspaces.