nLab regular representation




Many algebraic objects have a representation theory in which they act on other things; for example, groups often act on sets and algebras act on modules. In these situations, the acting object can often be viewed as an example of the type of thing acted upon: a group has an underlying set and an algebra has an underlying module. The multiplication in the acting object then defines an action of the object on this unstructured copy of itself. This is called the regular representation and is an extremely useful representation to study as it only involves the object itself, whence is in a sense canonical, but contains a lot of information, as opposed to, say, the trivial representation (which is also canonical).


The definition is the same in each case, but we shall give the actual definitions for the usual suspects.


Let GG be a group with multiplication μ\mu. Let us write |G|{|G|} for the underlying set of GG. The left regular representation of GG is

  1. as a permutation representation: the action G×|G||G|G \times {|G|} \to {|G|} defined by gh=μ(g,h)g \cdot h = \mu(g,h);

  2. as a linear representation: the corresponding representation on the linear span of GG.

The right regular representation is defined analogously.


Let AA be an associative unital algebra with multiplication μ\mu. Let us write |A|{|A|} for the underlying module of AA. The left regular representation of AA is the action A|A||A|A \otimes {|A|} \to {|A|} defined by am=μ(a,m)a \cdot m = \mu(a,m).

The right regular representation is defined analogously.

These can be seen as examples of a more general concept.


Let (C,,I)(C,\otimes,I) be a monoidal category. Let M=(|M|,μ,η)M = ({|M|},\mu,\eta) be a monoid in CC, where |M|{|M|} is the underlying object of MM in CC. The regular representation of MM is the action of MM on |M|{|M|} induced by the product μ\mu.


  • The regular representation of GG as a linear representation is the induced representation Ind 1 G1Ind_{1}^G 1 of the trivial representation along the inclusion of the trivial subgroup.


Over the complex numbers, the regular representation of a finite group is a direct sum that contains each irreducible representation ρ i\rho_i with multiplicity its dimension

[G]idim (ρ i)ρ i. \mathbb{C}[G] \;\simeq\; \underset{i}{\sum} dim_{\mathbb{C}}(\rho_i) \cdot \rho_i \,.

(e.g. tom Dieck 09, Thm. (1.10.2) and using Schur's lemma for the complex case)


Lecture notes include

Last revised on May 19, 2021 at 14:39:04. See the history of this page for a list of all contributions to it.