Polynomial identities for orbit numbers of general linear and unitary groups over finite fields Original Research Article

Let p(n) be the set of all partitions of n ε N and denote an element c = 1c12c2 … ncn ε p(n) by the sequence (c1, c2, … cn) ε Nn0 with ∑i = 1,…nci · I = n. For n ε N and set membership, variant ε {0, ± 1} we define image. Then fn, set membership, variant(q) equals the number of conjugacy classes in GLn(q) or Un(q2) for set membership, variant = 1 or −1 respectively or the number of adjoint GLn(q)- or Un(q2)-orbits on their finite Lie algebras, if set membership, variant = 0. In this paper we give a unified proof of this together with a polynomial identity for fn, set membership, variant(X), involving partitions and ‘multipartitions’ of n.