Room 217

Splitting Subspaces of Linear Transformations over Finite Fields

Samrith Ram

HRI

Let $m$, $n$ be positive integers and denote by $\mathbf F_q$ the finite field with
$q$ elements. Let $V$ be a vector space of dimension $mn$ over $\mathbf F_q$ and
$T : V -> V$ be a linear transformation. An $m$-dimensional subspace $W$
of $V$ is said to be $T$-splitting if
$$
V = W \oplus T W \oplus · · · \oplus T^{n-1} W.
$$
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.