The decomposition of permutation module for infinite Chevalley groups

gauss decomposition for chevalley groups, revisited

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...

Infinite Permutation Groups

Infinite Bounded Permutation Groups

This work concerns permutation groups on uncountable sets. The environment is ZFC, Set Theory with Axiom of Choice. Therefore the cardinal numbers are wellordered, a cardinal number k is infinite if and only if Ko < k, and each cardinal number k has a unique successor k. Throughout the paper Cl will be a set (usually infinite), n its cardinality, and m a cardinal number less than n. The support...

Aspects of infinite permutation groups

Until 1980, there was no such subgroup as ‘infinite permutation groups’, according to the Mathematics Subject Classification: permutation groups were assumed to be finite. There were a few papers, for example [10, 62], and a set of lecture notes by Wielandt [72], from the 1950s. Now, however, there are far more papers on the topic than can possibly be summarised in an article like this one. I s...

Infinite Dimensional Chevalley Groups and Kac-moody Groups over Z

Let A be a symmetrizable generalized Cartan matrix. Let g be the corresponding Kac-Moody algebra. Let G(R) be Tits’ Kac-Moody group functor over commutative rings R associated to g. Then G(R) is given by five axioms KMG1-KMG5 which are natural extensions of the properties of Chevalley-Demazure group schemes. We give a construction of the Tits functor for symmetrizable Kac-Moody groups G using i...