in the 1960's noboru iwahori and hideya matsumoto, eiichi abe and kazuo suzuki, and michael stein discovered that chevalley groups $g=g(phi,r)$ over a semilocal ring admit remarkable gauss decomposition $g=tuu^-u$, where $t=t(phi,r)$ is a split maximal torus, whereas $u=u(phi,r)$ and $u^-=u^-(phi,r)$ are unipotent radicals of two opposite borel subgroups $b=b(phi,r)$ and $b^-=b^-(phi,r)$ contai...

In this paper, we construct a q-deformation of the Witt-Burnside ring of a profinite group over a commutative ring, where q ranges over the ring of integers. When q = 1, this coincides with the Witt-Burnside ring introduced by A. Dress and C. Siebeneicher (Adv. Math. 70 (1988), 87-132). To achieve our goal we first show that there exists a q-deformation of the necklace ring of a profinite group...

Let K be a field and let G be a multiplicative group. The group ring K[G] is an easily defined, rather attractive algebraic object. As the name implies, its study is a meeting place for two essentially different algebraic disciplines. Indeed, group ring results frequently require a blend of group theoretic and ring theoretic techniques. A natural, but surprisingly elusive, group ring problem co...

A method is described by which it is possible to determine the high barrier to ring inversion and hindered rotation by neat racemization of optically active compounds when the racemization is caused by ring inversion or hindered rotation. The method is based preparation of sufficiently pure enantiomers, mainly by chromatography on swollen microcrystalline triacetylcellulose (TAC). By this t...

when addition and multiplication are defined in the obvious way, form a ring, the group-ring of G over K, which will be denoted by R (G, K). Henceforward, we suppose that K has the modulus 1, and we denote the identity in G by e0. Then R(G,K) has the modulus l.e0. Since no confusion can arise thereby, the element 1. e in R(G, K) will be written as e, and whenever it is convenient, the elements ...

Abstract Let R be a commutative ring with identity. Let G be a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if there is a maximal ideal of R containing both. In this paper we show that a ring R is finite if and only if clique number of the graph G (associated with R as above) is finite. We also have shown that for a semilocal ring R, the chr...

k — a regular commutative ring. (Main case of interest, as we shall see, is k = Z.) kG — the group ring over k of a group G. BG — the classifying space of G, a space with contractible universal cover and fundamental group G. Unique up to homotopy equivalence. Has the property that H∗(BG) is the Eilenberg-MacLane homology of G. K(R) — the (non-connective) K-theory spectrum of a ring R. If one ig...

Let KV ] G be the invariant ring of a nite linear group G GL(V), and let GU be the pointwise stabilizer of a subspace U V. We prove that the following numbers associated to the invariant ring decrease if one passes from KV ] G to KV ] G U : the minimal number of homogeneous generators, the maximal degree of the generators, the number of syzygies and other Betti numbers, the complete intersectio...

Let G be a finite group and let S be a G-set. The Burnside ring of G has a natural structure of a λ-ring, {λ}n∈N. However, a priori λn(S), where S is a G-set, can only be computed recursively, by first computing λ1(S), . . . , λn−1(S). In this paper we establish an explicit formula, expressing λn(S) as a linear combination of classes of G-sets.

In this chapter we give a categorical definition of the integral cohomology ring of a stack. For quotient stacks [X/G] the categorical cohomology ring may be identified with the equivariant cohomology H∗ G(X). Identifying the stack cohomology ring with equivariant cohomology allows us to prove that the cohomology ring of a quotient Deligne-Mumford stack is rationally isomorphic to the cohomolog...