Gerald Cliff

Units of integral group rings of metabelian groups

Abstract Let U1( Z G) denote the units of augmentation one of the integral group ring Z G of the finite group G. We prove that G has a normal torsion-free complement in U1( Z G) if G has an abelian normal subgroup A and G A is abelian, of odd order, or of exponent dividing 4 or 6.

Weil Representations of Symplectic Groups Over Rings

We are interested in Weil representations of Sp(2 n , R ), where R is the ring Z / p l Z , p is an odd prime and l is a positive integer, or, more generally, R = [Oscr ]/[pfr ] l , where [Oscr ] is the ring of integers of a local field, [pfr ] is the maximal ideal of [Oscr ] and [Oscr ]/[pfr ] has odd characteristic. One reason for this interest is that a continuous finite-dimensional complex representation of Sp(2 n , [Oscr ]) has to factor through a representation of Sp(2 n , [Oscr ]/[pfr ] l ) for some l .

Finite groups of matrices over group rings

We investigate certain finite subgroups r of GLn (Zf), where II is a finite nilpotent group. Such a group r gives rise to a Z[r x fl7-module; we study the characters of these modules to limit the structure of r. We also exhibit some exotic subgroups r.

On the trace of an idempotent in a group ring

Let KG be the group ring of a polycyclic by finite group G over a field K of characteristic zero. It is proved that if e = i e(g)g is a nontrivial idempotent in KG then its trace e(l) is a rational number r/s, (r, s) = 1, with the property that for every prime divisor p of s, G has an element of order p. This result is used to prove that if R is a commutative ring of characteristic zero, without nontrivial idempotents and G is a polycyclic by finite group such that no group order # 1 is invertible in R, then RG has no nontrivial idempotents. 1. Let KG be the group ring of a group G over a field K. By the trace of an element a = Eg a(g)g of KG is understood a(l), the coefficient of the identity in a. The following two statements regarding the trace of an idempotent in KG are well known. THEOREM (ZALESSKII [7]). The trace of an idempotent in KG lies in the prime subfield of K. THEOREM (KAPLANSKY, SEE [4]). If K is afield of characteristic zero, the trace of a nontrivial idempotent in KG lies strictly between 0 and 1. We expect that in the characteristic zero case one should be able to say more, namely the denominator of the trace of a nontrivial idempotent is a GI number, in the sense that for every prime p dividing this denominator, G has an element of order p. This statement is proved in Theorem 1 for polycyclic by finite groups. We apply this to prove that if R is a unital commutative ring of characteristic zero without nontrivial idempotents, with the property that no group element =# 1 has order invertible in R and G is polycyclic by finite, then RG has no nontrivial idempotents. This is proved for supersolvable groups in [3] and [5]. 2. Results. THEOREM 1. Let KG be the group ring of a polycyclic by finite group G over a field K of characteristic zero. Let e = ,g e(g)g be a nontrivial idempotent. Write Received by the editors March 20, 1976. AMS (MOS) subject classifications (1970). Primary 16A26; Secondary 20C05.

A Basis of Bideterminants for the Coordinate Ring of the Orthogonal Group

We give a basis of bideterminants for the coordinate ring K[O(n)] of the orthogonal group O(n,K), where K is an infinite field of characteristic not 2. The bideterminants are indexed by pairs of Young tableaux which are O(n)-standard in the sense of King–Welsh. We also give an explicit filtration of K[O(n)] as an O(n,K)-bimodule, whose factors are isomorphic to the tensor product of orthogonal analogs of left and right Schur modules.

Clifford and Mackey theory for Weil representations of symplectic groups

For a semi-simple group scheme G defined overZ, let X be an irreducible comple representation of the finite group G(R) whereR is Z/pZ, or more generallyR =O/P l whereP is the maximal ideal of the ringO of integers of a local field. One can try analyzeX using Clifford theory with respect to the congruence subgroup Γ (P ) which is the kernel of the natural map fromG(O/P ) to G(O/P i ), wherei is a positive intege less thanl. This involves finding an irreducible constituent Y of the restriction ofX to

Determining irreducible

We discuss several methods which yield spanning sets for the irreducible polynomial

GL(n,K)

GL(n,K)

-modules

-modules, where

Crystal Bases and Young Tableaux

K