2:00 pm, Lecture Hall 3 Splitting subspaces of linear transformations over finite fields Samrith Ram HRI, Allahabad. 030317 Abstract 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 mdimensional subspace W of V is said to be T splitting if V = W (+) T W (+) Â· Â· Â· (+) T^{n1} W. Determining the number of mdimensional Tsplitting 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 Tsplitting subspaces.
