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representation ring

Contents

Context

Representation theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

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.

As equivariant K-theory of the point

Equivalently the representation ring of GG over the complex numbers 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)KU G 0(*)KK(,C(BG)). R_{\mathbb{C}}(G) \simeq KU^0_G(\ast) \simeq KK(\mathbb{C}, C(\mathbf{B}G)) \,.

The first isomorphism here follows immediately from the elementary definition of equivariant topological K-theory, since a GG-equivariant vector bundle over the point is manifestly just a linear representation of GG on a complex vector space.

(e.g. Greenlees 05, section 3, Wilson 16, example 1.6 p. 3)

Therefore a similar isomorphism identifies the GG-representation ring over the real numbers with the equivariant orthogonal KK-theory of the point in degree 0:

R (G)KO G 0(*). R_{\mathbb{R}}(G) \;\simeq\; KO_G^0(\ast) \,.

But beware that equivariant KO, even of the point, is much richer in higher degree (Wilson 16, remark 3.34)

In fact, equivariant KO-theory of the point subsumes the representation rings over the real numbers, the complex numbers and the quaternions:

KO G n(*){0 | n=7 R (G)/R (G) | n=6 R (G)/R (G) | n=5 R (G)/R (G) | n=4 0 | n=3 R (G)/R (G) | n=2 R (G)/R (G) | n=1 R (G)/R (G) | n=0 KO_G^n(\ast) \;\simeq\; \left\{ \array{ 0 &\vert& n = 7 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{R}}(G) &\vert& n = 6 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 5 \\ R_{\mathbb{H}}(G) \phantom{/ R_{\mathbb{R}}(G) } &\vert& n = 4 \\ 0 &\vert& n = 3 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{H}}(G) &\vert& n = 2 \\ R_{\mathbb{R}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 1 \\ R_{\mathbb{R}}(G) \phantom{/ R_{\mathbb{R}}(G)} &\vert& n =0 } \right.

(Greenlees 05, p. 3)

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point *\ast
cohomology
of classifying space BGB G
(equivariant)
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
(equivariant)
complex K-theory
KUrepresentation ring
KU G(*)R (G)KU_G(\ast) \simeq R_{\mathbb{C}}(G)
Atiyah-Segal completion theorem
R(G)KU G(*)compl.KU G(*)^KU(BG)R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)
(equivariant)
complex cobordism cohomology
MUMU G(*)MU_G(\ast)completion theorem for complex cobordism cohomology
MU G(*)compl.MU G(*)^MU(BG)MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)
(equivariant)
algebraic K-theory
K𝔽 pK \mathbb{F}_prepresentation ring
(K𝔽 p) G(*)R p(G)(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)
Rector completion theorem
R 𝔽 p(G)K(𝔽 p) G(*)compl.(K𝔽 p) G(*)^Rector 73K𝔽 p(BG)R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)
(equivariant)
stable cohomotopy
K 𝔽 1Segal 74\mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} SBurnside ring
𝕊 G(*)A(G)\mathbb{S}_G(\ast) \simeq A(G)
Segal-Carlsson completion theorem
A(G)Segal 71𝕊 G(*)compl.𝕊 G(*)^Carlsson 84𝕊(BG)A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(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 ρtr(ρ()):G\chi_\rho \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 Schur’s lemma

The (isomorphism classes) of finite-dimensional irreducible representations V iV_i form the canonical \mathbb{Z}-linear basis of the representation ring:

R(G)[{V i} i]. R(G) \;\simeq\; \mathbb{Z}\big[ \{V_i\}_i \big] \,.

Moreover, the function assigning dimensions of hom-vector spaces constitutes a canonical bilinear symmetric inner product:

,:R(G)×R(G). \langle -,-\rangle \;\colon\; R(G) \times R(G) \longrightarrow \mathbb{Z} \,.

In terms of this, Schur's lemma states that the irreducible representations generally constitute an orthogonal basis, and even an orthonormal basis when the ground field is algebraically closed.

For more discussion of this perspective, see

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.

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point *\ast
cohomology
of classifying space BGB G
(equivariant)
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
(equivariant)
complex K-theory
KUrepresentation ring
KU G(*)R (G)KU_G(\ast) \simeq R_{\mathbb{C}}(G)
Atiyah-Segal completion theorem
R(G)KU G(*)compl.KU G(*)^KU(BG)R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)
(equivariant)
complex cobordism cohomology
MUMU G(*)MU_G(\ast)completion theorem for complex cobordism cohomology
MU G(*)compl.MU G(*)^MU(BG)MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)
(equivariant)
algebraic K-theory
K𝔽 pK \mathbb{F}_prepresentation ring
(K𝔽 p) G(*)R p(G)(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)
Rector completion theorem
R 𝔽 p(G)K(𝔽 p) G(*)compl.(K𝔽 p) G(*)^Rector 73K𝔽 p(BG)R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)
(equivariant)
stable cohomotopy
K 𝔽 1Segal 74\mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} SBurnside ring
𝕊 G(*)A(G)\mathbb{S}_G(\ast) \simeq A(G)
Segal-Carlsson completion theorem
A(G)Segal 71𝕊 G(*)compl.𝕊 G(*)^Carlsson 84𝕊(BG)A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)

Lambda-ring structure

The Adams operations equip the representation ring with the structure of a Lambda ring,. (e.g. tomDieck 79, section 3.5, tom Dieck 09, section 6.2, Meir 17).

Relation to the Burnside ring

Let GG be a finite group. Consider

  1. the Burnside ring A(G)A(G), which is the Grothendieck group of the monoidal category GSetG Set of finite G-sets;

  2. the representation ring R(G)R(G), which is the Grothendieck group of the monoidal category GRepG Rep of finite dimensional GG-linear representations.

Then then map that sends a G-set to the corresponding linear permutation representation is a strong monoidal functor

GSet[]GRep G Set \overset{\mathbb{C}[-]}{\longrightarrow} G Rep

and hence induces a ring homomorphism

A(G)[]R(G) A(G) \overset{ \mathbb{C}[-] }{\longrightarrow} R(G)

Under the identitification

  1. of the Burnside ring with the equivariant stable cohomotopy of the point

    A(G)𝕊 G(*) A(G) \;\simeq\; \mathbb{S}_G(\ast)

    (see there)

  2. of the representation ring with the equivariant K-theory of the point

    R(G)K G(*) R(G) \;\simeq\; K_G(\ast)

    (as above)

this should be image of the initial morphism of E-infinity ring spectra

𝕊KU \mathbb{S} \longrightarrow KU

from the sphere spectrum to KU.

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

General

Lecture notes include

Exposition in relation to equivariant K-theory includes

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.

Adams operations

The Adams operations and Lambda-ring-structure on representation rings are discussed in

Last revised on October 2, 2018 at 12:13:05. See the history of this page for a list of all contributions to it.