Group actions on Cartesian powers with applications to representation theory. (English) Zbl 0613.20008

Pahlings, H.; Plesken, Wilhelm

J. Reine Angew. Math. 380, 178-195 (1987).

The diagonal operation of a (finite) group G on the Cartesian powers \(V^ n\) (n\(\in N)\) of a G-set V defines a set of subgroups closed under conjugation and taking intersectons, namely the set of stabilizers of elements of \(V^ n\) for some n. A closure operation is defined on the lattice of all subgroups of G such that these stabilizers and G are exactly the closed subgroups. The closed subgroups are in Galois correspondence with their fixed points in V. There are many contexts in which these closed subgroups turn up naturally, e.g. Galois theory, Young subgroups of symmetric groups, Bravais groups in crystallography, and as inertia factors in Clifford theory. Any set of subgroups of a finite group G closed under conjugation and taking intersections can be realized as a set of closed subgroups with respect to a unique minimal G-set V.
For the case that V is a G-module rational polynomials are defined, which count the G-orbits of given type in \(V^ n\) for finite V and all n. This idea is modified to G-modules in characteristic 0 and for \({\mathbb{Z}}G\)- lattices. For some modules of certain groups these polynomials factorize completely over the rationals, e.g. for reflection modules or natural modules of classical groups over finite fields. In fact in the context of reflection groups similar polynomials with different interpretations turned up in the work of L. Solomon and P. Orlik.
Finally these ideas are applied to study conjugacy classes and ordinary or projective complex characters of infinite series of extensions \(E_ n\) of abelian groups \(W^ n\) by a fixed finite group G. The G-module W might be finite, free abelian or free over the p-adic integers. All characters are assumed to factor over some finite factor groups. B. Fischer’s concept of Clifford matrices is used and generic characters and conjugacy classes, and generic Clifford matrices are defined, which together with the character tables of the inertia factors \((=\) closed subgroups of G) yield the character tables of each \(E_ n\) after suitable specialization. Most generic computations to be carried out just amount to (properly done) calculations in \(E_ 1\), but suitably interpreted.

MSC:

20C15	Ordinary representations and characters
20C25	Projective representations and multipliers
06A15	Galois correspondences, closure operators (in relation to ordered sets)
20C05	Group rings of finite groups and their modules (group-theoretic aspects)
20D30	Series and lattices of subgroups

Keywords:

lattice of subgroups; diagonal operation; Cartesian powers; stabilizers; closure operation; closed subgroups; Galois correspondence; fixed points; inertia factors; Clifford theory; G-orbits; reflection groups; conjugacy classes; projective complex characters; generic characters; generic Clifford matrices; character tables