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?
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Any group $G$ has a category of finite-dimensional complex-linear representations, often denoted $Rep(G)$. This is a symmetric monoidal abelian category (a “tensor category”) and thus has a Grothendieck ring, which is called the representation ring of $G$ and denoted $R(G)$. Elements of the representation ring are hence formal differences (with respect to direct sum) of ordinary representations: virtual representations.
More concretely, we get $R(G)$ as follows. It has a basis $(e_i)_i$ given by the isomorphism classes of irreducible representations of $G$: that is, $i$ is an index for an irreducible finite-dimensional complex representation of $G$. It has a product given by
where $m_{i j}^k$ is the multiplicity of the $k$th irrep in the tensor product of representations of the $i$th and $j$th irreps.
In physics these coefficients are also known as Clebsch-Gordan coefficients (specifically for $G$ the special orthogonal group $SO(3)$), and the relations they satisfy are also known as Fierz identities (specifically for $G$ a spin group).
Notice that $R(G)$ is commutative thanks to the symmetry of the tensor product.
Equivalently the representation ring of $G$ is the $G$-equivariant K-theory of the point, or equivalently by the Green-Julg theorem, if $G$ is a compact Lie group, the operator K-theory of the group algebra (the groupoid convolution algebra of the delooping groupoid of $G$):
If $G$ is a finite group and we tensor $R(G)$ with the complex numbers, it becomes isomorphic to the character ring of $G$: that is, the ring of complex-valued functions on $G$ that are constant on each conjugacy class. Such functions are called class functions.
Similarly for $G$ a compact Lie group, its complex linear representations $\rho \colon G \to U(n) \to Aut(\mathbb{C}^n)$ (for all $n \in \mathbb{N}$) are uniquely specified by their characters $\chi_\tho \coloneqq tr(\rho(-)) \colon G \to \mathbb{C}$. Therefore also here the representation ring is often called the character ring of the group.
The representation ring of a compact Lie group is equivalent to the $G$-equivariant K-theory of the point.
The construction of representations by index-constructions of $G$-equivariant Dirac operators (push-forward in $G$-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 $Rep(G)$ at the augmentation ideal appears
The completion of $Rep(Spin(2k))$ is $\mathbb{Z}[ [ e^{\pm x_j} ] ]$ for $1 \leq j \leq k$ (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:
Gregory Landweber, Representation rings of Lie superalgebras, K-Theory 36 (2005), no. 1-2, 115-168 (arXiv:math/0403203)
Gregory Landweber, Twisted representation rings and Dirac induction, J. Pure Appl. Algebra 206 (2006), no. 1-2, 21-54 (arXiv:math/0403524)
With an eye towards loop group representations: