geometric representation theory
representation, 2-representation, ∞-representation
group algebra, algebraic group, Lie algebra
vector space, n-vector space
affine space, symplectic vector space
module, equivariant object
bimodule, Morita equivalence
induced representation, Frobenius reciprocity
Hilbert space, Banach space, Fourier transform, functional analysis
orbit, coadjoint orbit, Killing form
geometric quantization, coherent state
module algebra, comodule algebra, Hopf action, measuring
Geometric representation theory
D-module, perverse sheaf,
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
geometric function theory, groupoidification
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Special and general types
Any group has a category of finite-dimensional complex-linear representations, often denoted . This is a symmetric monoidal abelian category (a “tensor category”) and thus has a Grothendieck ring, which is called the representation ring of and denoted . Elements of the representation ring are hence formal differences (with respect to direct sum) of ordinary representations: virtual representations.
More concretely, we get as follows. It has a basis given by the isomorphism classes of irreducible representations of : that is, is an index for an irreducible finite-dimensional complex representation of . It has a product given by
where is the multiplicity of the th irrep in the tensor product of representations of the th and th irreps.
In physics these coefficients are also known as Clebsch-Gordan coefficients (specifically for the special orthogonal group ), and the relations they satisfy are also known as Fierz identities (specifically for a spin group).
Notice that is commutative thanks to the symmetry of the tensor product.
In terms of K-theory
Equivalently the representation ring of is the -equivariant K-theory of the point, or equivalently by the Green-Julg theorem, if is a compact Lie group, the operator K-theory of the group algebra (the groupoid convolution algebra of the delooping groupoid of ):
Relation to the character ring
If is a finite group and we tensor with the complex numbers, it becomes isomorphic to the character ring of : that is, the ring of complex-valued functions on that are constant on each conjugacy class. Such functions are called class functions.
Similarly for a compact Lie group, its complex linear representations (for all ) are uniquely specified by their characters . Therefore also here the representation ring is often called the character ring of the group.
Relation to equivariant K-theory
The representation ring of a compact Lie group is equivalent to the -equivariant K-theory of the point.
The construction of representations by index-constructions of -equivariant Dirac operators (push-forward in -equivariant K-theory to the point) is called Dirac induction.
On the other hand, by the Atiyah-Segal completion theorem in Borel-equivriant K-theory only the completion of at the augmentation ideal appears
The completion of is for (e.g Brylinski 90, p. 9).
Classical results for compact Lie groups:
Graeme Segal, The representation ring of a compact Lie group, Publications Mathématiques de l’Institut des Hautes Études Scientifiques January 1968, Volume 34, Issue 1, pp 113-128 (NUMDAM)
Masaru Tackeuchi, A remark on the character ring of a compact Lie group, J. Math. Soc. Japan Volume 23, Number 4 (1971), 555-705 (Euclid)
In the generality of super Lie groups:
With an eye towards loop group representations:
- Jean-Luc Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology, 29(4):461–480, 1990.