Grothendieck group

Summary

In mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in the most universal way, in the sense that any abelian group containing a homomorphic of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in , introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of of an , with the direct sum as its operation. Grothendieck group of a commutative monoid Motivation Given a commutative monoid M, "the most general" abelian group K that arises from M is to be constructed by introducing inverse elements to all elements of M. Such an abelian group K always exists; it is called the

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https://en.wikipedia.org/wiki/Grothendieck_group

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Study the basics of representation theory of groups and associative algebras.

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The composition factors of the functor of permutation modules

Mélanie Baumann

The Grothendieck group of permutation modules for a finite group G becomes a biset functor when G is allowed to vary. All the composition factors of this biset functor are determined.

Elsevier2011

Let k be an algebraically closed field of characteristic p, where p is a prime number or 0. Let G be a finite group and ppk(G) be the Grothendieck group of p-permutation kG-modules. If we tensor it with C, then Cppk becomes a C-linear biset functor. Recall that the simple biset functor SH,V are parametrized by pairs (H,V), where H is a finite group and V a simple COut(H)-module. If we only consider p'-groups, then Cppk = CRk is the usual representation functor and we know the simple functors which are its composition factors. If we consider only p-groups, then Cppk = CB is the Burnside functor and we also know the simple functors which are its composition factors. We want to find the composition factors of Cppk in general. In order to achieve this, we first show that the composition factors from the special cases above are also composition factors for Cppk. Then, we consider groups of little order and try to find new composition factors. This leads us to find the following new composition factors : The simple factors SCm,Cξ and SCp×Cp× Cm,Cξ, where (m,ξ) runs over the set of all pairs formed by a positive integer m prime to p and a primitive character ξ : (Z/mZ)* → C*. Their multiplicity as composition factors is 1. The simple factors SCp⋊Cl, C, where l is a number prime to p, the action of Cl on Cp is faithful and C is the trivial COut(Cp ⋊ Cl)-module. Their multiplicity as composition factors is φ(l). The simple functors SG,C, where G is a finite p-hypo-elementary B-group (for which an explicit classification is done) and C the trivial COut(G)-module. We also show that some specific simple functors appear, indexed by the groups C3 ⋊ C4, C5 ⋊ C4 and A4. On the way, we find all the composition factors of the subfunctor of permutation modules.

EPFL2012

Let k be a field of characteristic p, where p is a prime number, let pp_k(G) be the Grothendieck group of p-permutation kG-modules, where G is a finite group, and let Cpp_k(G) be pp_k(G) tensored with the field of complex numbers C. In this article, we find all the composition factors of the biset functor Cpp_k restricted to the category of abelian groups.

Elsevier2013