Let be a group and a field. Consider the class
where is a fixed vector space with a basis which is in bijective correspondence with . If is a required bijection, then we define ,,” on basis by
for some ( is a ,,free” representation). In particular every finite-dimensional representation is a quotient of a direct sum of copies of . This fact shows that a maximal subclass consisting of pairwise nonisomorphic representations is actually a set (note that is never unique). Fix such a set.
Definition. The representation semiring of is defined as a triple , where is a maximal set of pairwise nonisomorphic representations taken from . Addition and multiplication are given by
where is a representation in isomorphic to the direct sum and
The representation ring is defined as the Grothendieck ring (http://planetmath.org/GrothendieckGroup) induced from . It can be shown that the definition does not depend on the choice of (in the sense that it always gives us naturally isomorphic rings).
It is convenient to forget about formal definition which includes the choice of and simply write elements of as isomorphism classes of representations . Thus every element in can be written as a formal difference . And we can write
|Date of creation||2013-03-22 19:19:02|
|Last modified on||2013-03-22 19:19:02|
|Last modified by||joking (16130)|