POLYNOMIAL-IDENTITIES FOR ORBIT NUMBERS OF GENERAL LINEAR AND UNITARYGROUPS OVER FINITE-FIELDS

Let P(n) be the set of all partitions of n epsilon N and denote an ele ment c = 1(c1)2(c2)...n(cn) epsilon P(n) by the sequence (c(1), c(2),. ..c(n)) epsilon N-0(n) with Sigma(i=1,...,n)c(i) . i = n. For n epsilo n N and epsilon epsilon {0, +/- 1} We define [GRAPHICS] Then F-n,F-e(q ) equals the number of conjugacy classes in GL(n)(q) or U-n(q(2)) for epsilon = 1 or -1 respectively or the number of adjoint GL(n)(q)- or U -n(q(2))-orbits on their finite Lie algebras, if epsilon = 0. In this paper we give a unified proof of this together with a polynomial ident ity for F-n,F-e(X), involving partitions and 'multipartitions' of n. ( C) Elsevier Science Inc., 1997.