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
The notion of representation is closely related to, or even identical, to that of action: some object $C$ that has a notion of composition is represented on some object $D$ that has a notion of composition. In this generality representation is just another word for functor (or potentially $\infty$-functor). But in practice the term representation is typically used in the context of representation theory, where attempt to study $C$ in terms of its representations in $D$, where $D$ is typically rather more familiar.
Most specifically, one studies representations of a group by linear endomorphisms of a vector space; that is, $C$ is (the delooping of) a group and $D$ is Vect. However, the typical tools in representation theory these days involve vast generalizations of the notion of a linear representation of a group; for instance, one studies D-modules on action groupoids $G //_{Ad}G$ and things like that. This may be thought of as studying representations with values in ∞-vector spaces.
The notion of a group of ‘operators’ was already being used in about 1832 in work by Galois, and others, but there was not a definition of an abstract group until 40 years later when Cayley wrote:
A group is defined by the law of composition of its members.
(see the article on The abstract group concept in the St Andrews History of Mathematics archive.) Groups as well behaved sets of functions were beginning to be well understood and used, for instance in Klein’s work on geometry. Cayley proved that every finite group could be realised as a group of permutations. The theory of representations grew from that.
An abstract group could be studied by mapping it into a group of permutations or of invertible matrices, as then you could bring techniques form one area of mathematics (linear algebra) to the assistance of another, the Theory of Abstract Groups. This was exploited fully by Frobenius, Burnside, Schur and later Bauer.
That is one theme: you take an abstract algebraic thing and study it by mapping it into a similar structure which you think you know more about!
This also uses another basic idea from the start of group theory. The earlier pioneers thought of groups as groups of ‘operators’, but that means they have to operate on something. To make things more explicit, let $G$ be a finite group then we know there are homomorphisms from $G$ into $S_n$. (We could take $n$ to be the order of the group and use Cayley’s theorem but we do not assume that is the case.) If we look at such a homomorphism we get an action of $G$ on an $n$-element set.
Similarly if we take a homomorphism from $G$ into a group of invertible matrices $Gl_n(K)$, for $K$ a field, say, then we get a linear action of $G$ on the vector space, $K^n$.
As you would expect, we can generalise and categorify this basic idea in several useful ways.
We can think of $G$ as a groupoid, $\mathbf{B}G$, (the delooping of $G$), and then a linear representation / action will be a functor from $\mathbf{B}G$ to $Vect$, the category of vector spaces over $K$. We could replace $G$ by a general groupoid, or a general category, but then a representation of that is the same as a diagram of that ‘shape’ in $Vect$. We could replace $Vect$ by another more general category, or higher category, but if we are thinking of diagrams as representations, perhaps we should not totally forget that the term ‘representation’ did mean a process whereby the perhaps abstract ‘syntactical’ objects of the category gain a ‘semantic’ meaning, as ‘operations’ of some type, and which in turn, can be usefully used to gain information on the inherent structure.
In a rather general form, we therefore have a representation of a category $C$ in a category $D$ is simply a functor $F\colon C \to D$. Similarly, an homomorphism between representations (“intertwiner”) is simply a natural transformation between functors when they are being thought of as representations.
The term ‘representation’ is most often used when one or more of the following conditions apply:
The classical representation theory of groups is about representations of (finite, topological, smooth etc.) groups on (topological) vector spaces, that is when the first two conditions apply.
There are also enriched, $k$-linear and other versions, hence one can talk about representations of Lie algebras, vertex operator algebras, etc. See also representation theory.
representation, ∞-representation
equivariant homotopy theory, global equivariant homotopy theory
equivariant stable homotopy theory, global equivariant stable homotopy theory
representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:
homotopy type theory | representation theory |
---|---|
pointed connected context $\mathbf{B}G$ | ∞-group $G$ |
dependent type | ∞-action/∞-representation |
dependent sum along $\mathbf{B}G \to \ast$ | coinvariants/homotopy quotient |
context extension along $\mathbf{B}G \to \ast$ | trivial representation |
dependent product along $\mathbf{B}G \to \ast$ | homotopy invariants/∞-group cohomology |
dependent product of internal hom along $\mathbf{B}G \to \ast$ | equivariant cohomology |
dependent sum along $\mathbf{B}G \to \mathbf{B}H$ | induced representation |
context extension along $\mathbf{B}G \to \mathbf{B}H$ | |
dependent product along $\mathbf{B}G \to \mathbf{B}H$ | coinduced representation |
spectrum object in context $\mathbf{B}G$ | spectrum with G-action (naive G-spectrum) |