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representation ring∗
joking†
2013-03-22 3:41:40
Let G be a group and k a field. Consider the class
R = {X | X is a representation of G over k}
and its subclass Rf consisting of those representations which are finite-dimensional
as vector spaces. We consider a special representation
F = (V, ·)
where V is a fixed vector space with a basis B which is in bijective correspondence with G. If f : B → G is a required bijection, then we define ,,·” on basis
B by
g · 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 ∈ Rf , then there
exists an epimorphism of representations
e : Fn → X
for some n ∈ N (F is a ,,free” representation). In particular every finitedimensional representation is a quotient of a direct sum of copies of F. This fact
shows that a maximal subclass X ⊂ Rf consisting of pairwise nonisomorphic
representations is actually a set (note that X is never unique). Fix such a set.
Definition. The representation semiring Rk (G) of G is defined as a
triple (X , +, ·), where X is a maximal set of pairwise nonisomorphic representations taken from Rf . Addition and multiplication are given by
X +Y =Z
where Z is a representation in X isomorphic to the direct sum X ⊕ Y and
X · Y = Z0
∗ hRepresentationRingi
created: h2013-03-2i by: hjokingi version: h42255i Privacy setting: h1i hDefinitioni h20C99i
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where Z 0 is a representation in X isomorphic to the tensor product X ⊗Y . Note
that Rk (G) is not a ring, because there are no additive inverses.
The representation ring 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 difference [X]−[Y ].
And we can write
[X] + [Y ] = [X ⊕ Y ];
[X][Y ] = [X ⊗ Y ].
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