On the representations of \(q\) unipotent groups over a finite field.
Let \(q\) denote a power of a prime, and \(F_q\) denote the finite field with \(q\) elements. Let \(U_n(q)\) denote the finite group which consists of all upper triangular matrices of size \(n\) with ones along the diagonal. If \(U_n(q)\) has an irreducible representation of degree \(q^e\), then \(e\leq\mu(n)\), where \(\mu(n)=m(m-1)\) if \(n=2m\) and \(\mu(n)=m^2\) if \(n=2m+1\). Denote by \(N_{n,e}(q)\) the number of irreducible representations of degree \(q^e\). There are certain conjectures related to these numbers. In this paper, the author studies some conjectures related to the group \(U_n(q)\) and some of its subgroups. He shows that the numbers \(N_{n,e}(q)\) are all polynomials in \(q-1\). Concretely, in the first two sections he verifies the above conjecture for some \(q\) unipotent groups. In section 2 he proves it for the generalized Heisenberg groups, and for the maximal unipotent radicals of the classical groups \(\text{Sp}_6\) and \(\text{SO}_8\). In section 3 he verifies it also for the maximal unipotent subgroup of the exceptional group \(G_2\). In section 4 he tries to give such an interpretation for the group \(U_n\).
scientific article; zbMATH DE number 5884608
scientific article