Tuesday, February 28 2017
10:45 - 12:30

#### 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 foran arbitrary transformation $T$ is an open problem closely related tomany problems in combinatorics and cryptography. I will outline connectionswith a theorem of Philip Hall on conjugacy class size in thegeneral linear group and some results of Wilf et al. on the probabilityof coprime polynomials over finite fields. I will also discuss a generalenumeration problem on matrix polynomials which, if solved, wouldsettle the problem of counting $T$-splitting subspaces.

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