#### 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.

Done