nLab regular representation

Idea

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).

Definition

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

Definition

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

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

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

The right regular representation is defined analogously.

Definition

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

The right regular representation is defined analogously.

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

Definition

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

Properties

Proposition

The regular representation of $G$ as a linear representation is the induced representation $Ind_{1}^G 1$ of the trivial representation along the inclusion of the trivial subgroup.

Proposition

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

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

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

Taking dimensions on both sides of (1) yields the:

Proposition

(sum of squares formula)
The sum of the squares of the dimensions of the isomorphism classes of complex irreducible representations of a finite group $G$ equals the order ${\vert G \vert}$ of the group:

$$\sum_{ [\rho] \in Irrep(G)_{/\sim} } \big( dim \rho \big)^2 \;\; = \;\; {\vert G \vert} \,.$$

(cf. Etingof et al. 2011 Thm. 3.1(ii))

Related concepts

• trivial representation

• alternating representation

References

Most texts of representation theory discuss the regular representation, see there.

Monographs:

• Pavel Etingof, Dmitry Vaintrob et al.: Introduction to representation theory, Student Mathematical Library 59, AMS (2011) [arXiv:0901.0827, ams:stml-59]

  Lecture notes:

Lecture notes:

• Tammo tom Dieck, section 1.10 of Representation theory (2009) [pdf]