geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
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 $G$ be a group with multiplication $\mu$. Let us write ${|G|}$ for the underlying set of $G$. The left regular representation of $G$ is
as a permutation representation: the action $G \times {|G|} \to {|G|}$ defined by $g \cdot h = \mu(g,h)$;
as a linear representation: the corresponding representation on the linear span of $G$.
The right regular representation is defined analogously.
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.
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 $\mu$.
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
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.