Integral Representations of Finite Groups and Galois Stability
by
D.A. Malinin
Belarussian State Pedag. University

I present some arithmetic problems arising for representations of finite groups over arithmetic rings under the ground ring extensions.

1. The finiteness problem for the number of pairwise nonequivalent absolutely irreducible representations over certain rings is discussed. Some explicit constructions of infinite series for representations over the ring of integers of the field Q_p adjoined by all primitive p^m-roots r_p^m of 1 are given.

2. For a given number field F I consider [3], [4] a normal extension E/F of finite degree d and finite abelian subgroups G of GL_n(E) of a given exponent t. It is assumed that G is a finite linear group stable under the natural action of the Galois group T of E/F. Consider fields E=F(G) that are obtained via adjoining all matrix coefficients of all matrices g of G to F. It is proved that under some reasonable restrictions for n any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.

3. Let E/Q be a finite Galois extension of the rationals, T its Galois group, O_E the maximal order of E and G GL_n(O_E) is a finite T-stable subgroup of GL_n(O_E). Then there is the following

Conjecture 1. If E is totally real, then G is contained in GL_n(Z).

If E/Q is not totally real conjecture 1 can be generalized:

Conjecture 2. For any finite T-stable subgroup G of GL_n(O_E) the field Q(G) coincides with some cyclotomic subfield of E.

The following results are discussed in [2], [3].

1) Both conjectures are true in the case of Galois field extension E/Q with odd discriminant. Also some partial answers are given in the case of field extensions E/Q that are ramified at 2.

2) Conjecture 1 is true for totally real Galois extensions E/Q of degree [E:Q] <= 480 and for E/Q with [E:Q] <= 960 if we assume the generalized Riemann hypothesis. Conjecture 2 is true if [E:Q] < 288.

3) If all totally real fields Q(r2m+ r2m-1) have class number 1, then conjecture 1 holds for arbitrary solvable extensions E/Q.

Some related problems of globally irreducible representations of finite groups in the sense of F. Van Oystaeyen and A.E. Zalesskii [1] are also considered.

The interplay between the mentioned problems and finite arithmetic groups, integral representations, quadratic forms, Galois cohomology and Schur groups is discussed. Some links and applications are given.

[1] F. Van Oystaeyen; A.E. Zalesskii. "Finite groups over arithmetical rings and globally irreducible representations". J. Algebra 215, No.2, 418-436

[2] H.-J. Bartels, D. A. Malinin "Finite Galois stable subgroups of GL_n". Manuscripte der Forschergruppe Arithmetik, Mannheim-Heidelberg, No. 3 (2000), 21 S.

[3] D. A. Malinin "Galois stability for integral representations of finite groups". Algebra i analiz v. 12 (2000), No. 3, 106 - 142.

[4] D. A. Malinin "Galois stability for integral representations of finite abelian groups". Preprint of the Max-Planck-Institut No. 99-128 (1999), 20 S.

Date received: June 30, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafe-27.