# representation ring

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

 $\mathcal{R}=\{X\ |\ X\mbox{ is a representation of }G\mbox{ over }k\}$

and its subclass $\mathcal{R}_{f}$ consisting of those representations which are finite-dimensional as vector spaces. We consider a special representation

 $\mathcal{F}=(V,\cdot)$

where $V$ is a fixed vector space with a basis $\mathcal{B}$ which is in bijective correspondence with $G$. If $f:\mathcal{B}\to G$ is a required bijection, then we define ,,$\cdot$” on basis $\mathcal{B}$ by

 $g\cdot b=gf(b)$

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 $X\in\mathcal{R}_{f}$, then there exists an epimorphism of representations

 $e:\mathcal{F}^{n}\to X$

for some $n\in\mathbb{N}$ ($\mathcal{F}$ is a ,,free” representation). In particular every finite-dimensional representation is a quotient of a direct sum of copies of $\mathcal{F}$. This fact shows that a maximal subclass $\mathcal{X}\subset\mathcal{R}_{f}$ consisting of pairwise nonisomorphic representations is actually a set (note that $\mathcal{X}$ is never unique). Fix such a set.

Definition. The representation semiring $\overline{R_{k}(G)}$ of $G$ is defined as a triple $(\mathcal{X},+,\cdot)$, where $\mathcal{X}$ is a maximal set of pairwise nonisomorphic representations taken from $\mathcal{R}_{f}$. Addition and multiplication are given by

 $X+Y=Z$

where $Z$ is a representation in $\mathcal{X}$ isomorphic to the direct sum $X\oplus Y$ and

 $X\cdot Y=Z^{\prime}$

where $Z^{\prime}$ is a representation in $\mathcal{X}$ isomorphic to the tensor product $X\otimes Y$. Note that $\overline{R_{k}(G)}$ is not a ring, because there are no additive inverses.

$R_{k}(G)$ is defined as the Grothendieck ring (http://planetmath.org/GrothendieckGroup) induced from $\overline{R_{k}(G)}$. It can be shown that the definition does not depend on the choice of $\mathcal{X}$ (in the sense that it always gives us naturally isomorphic rings).

It is convenient to forget about formal definition which includes the choice of $\mathcal{X}$ and simply write elements of $\overline{R_{k}(G)}$ as isomorphism classes of representations $[X]$. Thus every element in $R_{k}(G)$ can be written as a formal difference $[X]-[Y]$. And we can write

 $[X]+[Y]=[X\oplus Y];$
 $[X][Y]=[X\otimes Y].$
Title representation ring RepresentationRing 2013-03-22 19:19:02 2013-03-22 19:19:02 joking (16130) joking (16130) 7 joking (16130) Definition msc 20C99