SEMINAR: Groups and Combinatorics Seminar:Primary Cyclic Matrices

Groups and Combinatorics Seminar

Brian Corr (UWA)

will speak on

Primary Cyclic Matrices

at 1pm Tuesday 7 September in MLR2

Abstract: Let G be an algebra of dxd matrices over a field F and denote m_X(t) and c_X(t) the minimal and the characteristic polynomials of X in G. X in called f-primary cyclic if f is a monic irreducible divisor if m_X(t) does not divide the quotient c_X(t)/m_X(t), and simply primary cyclic if it is f-primary cyclic for some f.

In 2006, Glasby described a version of the MEAT-AXE test for irreducibility which makes use of primary cyclic matrices, and in 2008 Glasby and Praeger determined the proportion of primary cyclic matrices in the full matrix algebra M(d,q). We use methods of generating functions and the Cycle Index Theorem to produce a lower bound for the proportion of primary cyclic matrices in the irreducible subalgebra M(c,q^b), and describe some asymptotic behaviour.

All welcome