nLab representation ring

Definition

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.

In components

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

$$e_i e_j = \sum_k m_{i j}^k e_k ,$$

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.

As equivariant K-theory of the point

Equivalently the representation ring of $G$ over the complex numbers 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 $G$):

$$R_{\mathbb{C}}(G) \simeq KU^0_G(\ast) \simeq KK(\mathbb{C}, C(\mathbf{B}G)) \,.$$

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

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

Therefore a similar isomorphism identifies the $G$-representation ring over the real numbers with the equivariant orthogonal $K$-theory of the point in degree 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:

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

(Greenlees 05, p. 3)

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

Properties

Relation to the character ring

If $G$ 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 $\chi_\rho \coloneqq tr(\rho(-)) \colon G \to \mathbb{C}$. 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 $V_i$ form the canonical $\mathbb{Z}$-linear basis of the representation ring:

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

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

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

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

• at Schur's lemma the section In terms of categorical algebra;

• at Gram-Schmidt process the section Categorified Gram-Schmidt process.

Relation to equivariant K-theory

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

$$Rep(G) \simeq K_G(\ast) \,.$$

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.

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

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 $G$ be a finite group. Consider

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

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

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

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

and hence induces a ring homomorphism

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

Under the identitification

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

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

   (see there)

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

   $$R(G) \;\simeq\; K_G(\ast)$$

   (as above)

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

$$\mathbb{S} \longrightarrow KU$$

from the sphere spectrum to KU.

f

Splitting principle and Brauer induction theorem

The Brauer induction theorem says that, over the complex numbers, the representation ring is generated already from the induced representations of 1-dimensional representations. This may be regarded as the splitting principle for linear representations and for characteristic classes of linear representations (Symonds 91).

Examples

Example

(representation ring of unitary groups) For $n \in \mathbb{N}$, the complex representation ring of U(n) is

$$R_{\mathbb{C}} \big( U(n) \big) \;\simeq\; \mathbb{Z}[x_1, \cdots, x_{n-1}, x_n, x_n^{-1}] \,,$$

and that of SU(n) is

$$R_{\mathbb{C}} \big( SU(n) \big) \;\simeq\; \mathbb{Z}[x_1, \cdots, x_{n-1}] \,.$$

(e.g. Husemöller 66, Ch. 14, Thm. 3.1 (p. 192))

• 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).

Related concepts

• Clebsch-Gordan coefficients, Fierz identities

• Deligne's theorem on tensor categories

References

General

Lecture notes:

• Tammo tom Dieck, Section 4.4. in: Representation theory, 2009 (pdf)

Computations for the classical Lie groups:

• Dale Husemöller, Representation Rings of Classical Groups, Chapter 14 in: Fibre bundles, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. 20, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) (pdf)

Exposition in relation to equivariant K-theory includes

• Akhil Mathew, Equivariant K-theory

• John Greenlees, Equivariant version of real and complex connective K-theory, Homology Homotopy Appl. Volume 7, Number 3 (2005), 63-82. (Euclid:1139839291)

• Dylan Wilson, Equivariant K-theory, 2016 (pdf, pdf)

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:PMIHES_1968__34__113_0)

• 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:

• Jean-Luc Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology, 29(4):461–480, 1990.

Concerning the Brauer induction theorem and the splitting principle:

• Peter Symonds, A splitting principle for group representations, Comment. Math. Helv. (1991) 66: 169 (dml:140229, doi:10.1007/BF02566643)

Adams operations

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

• Tammo tom Dieck, section 3.5 of Transformation Groups and Representation Theory, Lecture Notes in Mathematics 766 Springer 1979

• Robert Boltje, A characterization of Adams operations on representation rings, 2001 (pdf)

• Tammo tom Dieck, section 6.2 of Representation theory, 2009 (pdf)

• Michael Boardman, Adams operations on Group representations, 2007 (pdf)

• Ehud Meir, Markus Szymik, Adams operations and symmetries of representation categories (arXiv:1704.03389)