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].