nLab linear representation

Idea

A linear representation is a representation on a category of vector spaces or similar (Vect, Mod, etc.)

This is the most common flavor of representations. One sometimes considers representations on objects other than linear spaces (such as permutation representations) but often these are called not representations but actions.

Properties

Characters of linear representations

See characters of linear representations.

Characteristic classes of linear representations

Under the Atiyah-Segal completion map linear representations of a group $G$ induce topological K-theory classes on the classifying space $B G$. Their Chern classes are hence invariants of the linear representations themselves.

See at characteristic class of a linear representation for more.

Related concepts

• permutation representation

• category of representations

References

• Tammo tom Dieck, Representation theory, 2009 (pdf)

For more see the references at representation theory.