Separable exterior squares over finite fields

Abstract

The paper concerns exterior squares of polynomials and matrices over the finite field F q for large q. We find the probability that monic f∈ F q[t] has a non-separable exterior square. We then find the probability that X∈GL( d, q) has an exterior square which is non-separable, non-cyclic or non-semisimple. This should have applications in recognising GL( V) in its action on V∧ V, when V is a d-dimensional vector space over F q .