Polynomial identities for orbit numbers of general linear and unitary groups over finite fields (Q677146)

Let \(n\in N\) and \(P(n)\) be the set of all partitions of the number \(n\). Denote an element \(c=1^{c_1}\cdot 2^{c_2}\cdots n^{c_n}\) of \(P(n)\) by the sequence \((c_1,c_2,\dots,c_n)\in N_0^n\) with \(\sum_{i=1,2,\dots,n} c_i\cdot i=n\). For \(n\in N\) and \(\varepsilon\in\{0,1,-1\}\) consider the polynomial \(F_{n,\varepsilon}(X)=\sum_{c_k\in P(n)}\prod_{k=1,2,\dots,n, c_k>0} X^{c_k-1}(X-\varepsilon)\). Then \(F_{n,1}(q)\) is equal to the number of conjugacy classes in \(\text{GL}_n(q)\) over a finite field with \(q\) elements, \(F_{n,-1}(q)\) is equal to the number of conjugacy classes in \(U_n(q^2)\), and \(F_{n,0}(q)\) gives the number of \(\text{GL}_n(q)\) orbits of matrices in \(\text{Mat}_n(F_q)\) along with the number of adjoint \(U_n(q^2)\)-orbits on its Lie algebra \(U_n(q^2)=\{m\in\text{Mat}_n(F_{q^2}):m=(-m^{tr})^{(q)}\}\), where \(m^{tr}\) denotes the transpose of \(m\) and \(m^{(q)}\) the matrix obtained from \(m\) by raising each entry to the \(q\)th power. By \(CL_{n,\varepsilon}(q)\) we shall mean the number of orbits of matrices in the various afore-mentioned cases. \textit{G. E. Wall} [J. Aust. Math. Soc. 3, 1-62 (1963; Zbl 0122.28102) and Bull. Aust. Math. Soc. 22, 339-364 (1980; Zbl 0438.20031)]\ or \textit{I. G. Macdonald} [Bull. Aust. Math. Soc. 23, 23-48 (1981; Zbl 0445.20029)]\ have established the formula \(CL_{n,\varepsilon}(q)=F_{n,\varepsilon}(q)\) for these conjugacy classes. The author of the paper under review establishes this formula for Lie algebras. He also obtains partitions and multipartitions of \(n\).