Old and new in the supercharacter theory of finite groups

A. N. Panov

Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 227-249.

Résumé

The problem of classification of irreducible representations is a very complicated, "wild" problem for some groups like maximal unipotent, Borel and parabolic subgroups of the finite simple groups of Lie type. In 1962, A. A. Kirillov discovered the orbit method that establishes a one to one correspondence between the irreducible representations of a nilpotent Lie group and the coadjoit orbits. In 1977, D. Kazhdan modified the orbit method to be true for finite unipotent groups. However, the orbit method does not solve the problem, since the problem of classidication of the coadjpit orbits is a "wild" problem again. In 1995–2003, C. Andre constructer the theory of basic characters for the unitriangular group $\mathrm{UT}(n,{\mathbb F}_q)$. These characters are not irreducible, but they have many common features with the irreducible characters. The Andre theory was simplified be Ning Yan in 2003. In 2008, P. Diaconis and I. Isaacs formulated the general notion of a supercharacter theory and constructed the supercharacter theory for algebra groups, its precial case is the Andre theory of basic characters. The general problem is to construct for a given group a supercharacter theory that as close to the theory of irreducible characters as possible. Many papers were devoted to the supercharacter theory. Up today the case of abelian groups is studied in details; the connection with Gauss, Kloosterman and Ramanujan sums is investigated. The supercharacter theories for maximal unipotent subgroups in orthogonal and symplectic groups are constructed. The problems of restriction and superinduction is solved for the basic characters. The problem of construction of a supercharacter theory for the parabolic subgroups is still open. In §1–2 of the present paper, we present the authors proof of the main statements of the supercharacter theory for algebra groups, following the context of the paper of P. Diaconis and I. Isaacs. In §3, we announce the authors results on the supercharacter theory for the finite groups of triangular type, for which the theory of P. Diaconis and I.Isaacsas is a special case. We obtain the analog of A. A. Kirillov formula for irreducible characters. We show that the restriction of the supercaracter on a subgroup of triangular type is a sum supercharacters of these subgroup. As in the case of algebra group, the induction does not work for supercharacters. We defined a superinduction, obeying the main properties of induction including the Frobenius formula. Bibliography: 28 titles.

Détail

Keywords: supercharacter theory, algebra group, group representations, triangular group.

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