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representation ring

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Definition

Any group GG has a category of finite-dimensional complex-linear representations, often denoted Rep(G)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 GG and denoted R(G)R(G). Elements of the representation ring are hence formal differences (with respect to direct sum) of ordinary representations: virtual representations.

In components

More concretely, we get R(G)R(G) as follows. It has a basis (e i) i(e_i)_i given by the isomorphism classes of irreducible representations of GG: that is, ii is an index for an irreducible finite-dimensional complex representation of GG. It has a product given by

e ie j= km ij ke k, e_i e_j = \sum_k m_{i j}^k e_k ,

where m ij km_{i j}^k is the multiplicity of the kkth irrep in the tensor product of representations of the iith and jjth irreps.

In physics these coefficients are also known as Clebsch-Gordan coefficients (specifically for GG the special orthogonal group SO(3)SO(3)), and the relations they satisfy are also known as Fierz identities (specifically for GG a spin group).

Notice that R(G)R(G) is commutative thanks to the symmetry of the tensor product.

In terms of K-theory

Equivalently the representation ring of GG is the GG-equivariant K-theory of the point, or equivalently by the Green-Julg theorem, if GG is a compact Lie group, the operator K-theory of the group algebra (the groupoid convolution algebra of the delooping groupoid of GG):

R(G)K G(*)KK(,C(BG)). R(G) \simeq K_G(\ast) \simeq KK(\mathbb{C}, C(\mathbf{B}G)) \,.

Properties

Relation to the character ring

If GG is a finite group and we tensor R(G)R(G) with the complex numbers, it becomes isomorphic to the character ring of GG: that is, the ring of complex-valued functions on GG that are constant on each conjugacy class. Such functions are called class functions.

Similarly for GG a compact Lie group, its complex linear representations ρ:GU(n)Aut( n)\rho \colon G \to U(n) \to Aut(\mathbb{C}^n) (for all nn \in \mathbb{N}) are uniquely specified by their characters χ thotr(ρ()):G\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.

Relation to equivariant K-theory

The representation ring of a compact Lie group is equivalent to the GG-equivariant K-theory of the point.

Rep(G)K G(*). Rep(G) \simeq K_G(\ast) \,.

The construction of representations by index-constructions of GG-equivariant Dirac operators (push-forward in GG-equivariant K-theory to the point) is called Dirac induction.

Completion

On the other hand, by the Atiyah-Segal completion theorem in Borel-equivriant K-theory only the completion of Rep(G)Rep(G) at the augmentation ideal appears

Examples

Spin group

The completion of Rep(Spin(2k))Rep(Spin(2k)) is [[e ±x j]]\mathbb{Z}[ [ e^{\pm x_j} ] ] for 1jk1 \leq j \leq k (e.g Brylinski 90, p. 9).

References

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.

Revised on January 10, 2017 05:34:25 by Urs Schreiber (188.100.201.15)