representation ring

Let G be a group and k a field. Consider the class

={X|X is a representation of G over k}

and its subclass f consisting of those representationsPlanetmathPlanetmath which are finite-dimensional as vector spacesMathworldPlanetmath. We consider a special representation


where V is a fixed vector space with a basis which is in bijectiveMathworldPlanetmathPlanetmath correspondence with G. If f:G is a required bijection, then we define ,,” on basis by


where on the right side we have a multiplication in G. It can be shown that this gives us a well-defined representation and further more, if Xf, then there exists an epimorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of representations


for some n ( is a ,,free” representation). In particular every finite-dimensional representation is a quotient of a direct sumMathworldPlanetmathPlanetmathPlanetmathPlanetmath of copies of . This fact shows that a maximal subclass 𝒳f consisting of pairwise nonisomorphic representations is actually a set (note that 𝒳 is never unique). Fix such a set.

Definition. The representation semiringMathworldPlanetmath Rk(G)¯ of G is defined as a triple (𝒳,+,), where 𝒳 is a maximal set of pairwise nonisomorphic representations taken from f. Addition and multiplication are given by


where Z is a representation in 𝒳 isomorphic to the direct sum XY and


where Z is a representation in 𝒳 isomorphic to the tensor productPlanetmathPlanetmathPlanetmath XY. Note that Rk(G)¯ is not a ring, because there are no additive inverses.

The representation ringMathworldPlanetmath Rk(G) is defined as the Grothendieck ring ( induced from Rk(G)¯. It can be shown that the definition does not depend on the choice of 𝒳 (in the sense that it always gives us naturally isomorphic rings).

It is convenient to forget about formal definition which includes the choice of 𝒳 and simply write elements of Rk(G)¯ as isomorphism classes of representations [X]. Thus every element in Rk(G) can be written as a formal differencePlanetmathPlanetmath [X]-[Y]. And we can write

Title representation ring
Canonical name RepresentationRing
Date of creation 2013-03-22 19:19:02
Last modified on 2013-03-22 19:19:02
Owner joking (16130)
Last modified by joking (16130)
Numerical id 7
Author joking (16130)
Entry type Definition
Classification msc 20C99