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representation ring

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Representation theory

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cohomology

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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 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 irreps 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 the iith and jjth irreps. Note 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 March 29, 2014 08:54:04 by Urs Schreiber (89.204.138.114)