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.

The representation ring $R_{k}(G)$ 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
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