Primary cyclic matrices in irreducible matrix subalgebras

Abstract

Abstract Primary cyclic matrices were used (but not named) by Holt and Rees in their version of Parker’s MEAT-AXE algorithm to test irreducibility of finite matrix groups and algebras. They are matrices X with at least one cyclic component in the primary decomposition of the underlying vector space as an X-module. Let M ⁡ ( c , q b ) {\operatorname{M}(c,q^{b})} be an irreducible subalgebra of M ⁡ ( n , q ) {\operatorname{M}(n,q)} , where n = b ⁢ c > c {n=bc>c} . We prove a generalisation of the Kung–Stong cycle index theorem, and use it to obtain a lower bound for the proportion of primary cyclic matrices in M ⁡ ( c , q b ) {\operatorname{M}(c,q^{b})} . This extends work of Glasby and the second author on the case b = 1 {b=1} .