Presentation ring

The representation ring of a group is important in mathematics , especially in representation theory , but also in algebra , topology and K-theory .

definition

The representation ring of a group is defined as the Abelian group of the formal differences of representations, with direct sum and tensor product as addition and multiplication. ${\displaystyle G}$

For finite or compact groups , the representation ring can be defined equivalently as the Abelian group

${\displaystyle R(G)=\{\sum _{j=1}^{m}a_{j}\tau _{j}|a_{j}\in \mathbb {Z} ,\tau _{j},j=1,\dotsc ,m\,\,{\text{ alle}}{\text{ irreduziblen}}{\text{ Darstellungen}}{\text{ von }}G{\text{ über }}\mathbb {C} \,\,{\text{ bis}}{\text{ auf}}{\text{ Isomorphie}}\},}$

which becomes a ring with component-wise addition as well as that through the decomposition of the tensor product as a direct sum of irreducible representations as multiplication . The elements of are called virtual representations. ${\displaystyle R(G)}$

Operations

Direct sum

Be and two representations of a group . The direct sum of representations defines an addition ${\displaystyle (\rho _{1},V_{\rho _{1}})}$ ${\displaystyle (\rho _{2},V_{\rho _{2}})}$ ${\displaystyle G}$

${\displaystyle \left[\rho _{1}\right]+\left[\rho _{2}\right]:=\left[\rho _{1}\oplus \rho _{2}\right]}$

on . ${\displaystyle R(G)}$

Tensor product

Let be and two groups with respective representations and then a representation of the direct product is the tensor product of the two representations . That defines a homomorphism ${\displaystyle G_{1}}$ ${\displaystyle G_{2}}$ ${\displaystyle (\rho _{1},V_{\rho _{1}})}$ ${\displaystyle (\rho _{2},V_{\rho _{2}}),}$ ${\displaystyle \rho _{1}\otimes \rho _{2}}$ ${\displaystyle G_{1}\times G_{2}}$

${\displaystyle R(G_{1})\otimes _{\mathbb {Z} }R(G_{2})\to R(G_{1}\times G_{2}),}$

where the tensor product of the representation rings is as -modules. For , in particular, a multiplication is obtained by combining with the homomorphism defined by the diagonal embedding ${\displaystyle R(G_{1})\otimes _{\mathbb {Z} }R(G_{2})}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle G_{1}=G_{2}}$ ${\displaystyle G\to G\times G}$ ${\displaystyle R(G\times G)\to R(G)}$

${\displaystyle R(G)\otimes _{\mathbb {Z} }R(G)\to R(G)}$.

External product

For every representation of a group and every natural number one can define the -th outer product, which in turn is a representation of . This defines a sequence of operations ${\displaystyle G}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle G}$

${\displaystyle \lambda ^{n}\colon R(G)\to R(G)}$,

which make a λ-ring . ${\displaystyle R(G)}$

Adams operations

The Adams operations on the presentation ring of a compact group are defined by their effect on characters: ${\displaystyle \Psi ^{k}\colon R(G)\to R(G)}$

${\displaystyle \Psi ^{k}(\chi (g))=\chi (g^{k})}$.

They define ring homomorphisms and their effect on dimensional representations can be described by ${\displaystyle d}$

${\displaystyle \Psi ^{k}(\rho )=N_{k}(\Lambda ^{1}\rho ,\Lambda ^{2}\rho ,\ldots ,\Lambda ^{d}\rho )}$

where are the outer powers of and expresses the -th power sum as the sum of the elementary symmetric functions in variables. ${\displaystyle \Lambda ^{i}\rho }$ ${\displaystyle \rho }$ ${\displaystyle N_{k}}$ ${\displaystyle k}$ ${\displaystyle d}$

Examples

• For the cyclic group is ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$

${\displaystyle R(\mathbb {Z} /n\mathbb {Z} )=\mathbb {Z} \left[X\right]/(X^{n}-1)}$,

where corresponds to a 1-dimensional representation, which maps the producer of to a -th primitive unit root . ${\displaystyle X}$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ ${\displaystyle n}$

• For the symmetric group is ${\displaystyle S_{3}}$

${\displaystyle R(S_{3})=\mathbb {Z} \left[X,Y\right]/(XY-Y,X^{2}-1,Y^{2}-X-Y-1)}$,

where corresponds to the 1-dimensional alternating representation and the 2-dimensional irreducible representation of . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle S_{3}}$

• For the circle group is ${\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} }$

${\displaystyle R(S^{1})=\mathbb {Z} \left[X,X^{-1}\right]}$.

• For the special unitary group is ${\displaystyle SU(n)}$

${\displaystyle R(SU(n))=\mathbb {Z} \left[X_{1},\ldots ,X_{n}\right]/(X_{1}\ldots X_{n}-1)}$,

where corresponds to the representation that maps a diagonal matrix onto its -th diagonal entry. ${\displaystyle X_{i}}$ ${\displaystyle i}$

Presentation rings of compact groups

In the following we assume a compact (e.g. finite) group. ${\displaystyle G}$

Characters and representation rings

The character defines a homomorphism in the set of all class functions to complex-valued ${\displaystyle G}$

${\displaystyle {\begin{aligned}\chi :R(G)&\to \mathbb {C} _{\text{class}}(G)\\\sum a_{j}\tau _{j}&\mapsto \sum a_{j}\chi _{j},\end{aligned}}}$

where the corresponding irreducible characters are. ${\displaystyle \chi _{j}}$ ${\displaystyle \tau _{j}}$

For compact groups , a representation is determined by their character, therefore it is injective . The pictures of are called virtual characters. Since the irreducible characters form an orthonormal basis of , induces an isomorphism ${\displaystyle G}$ ${\displaystyle \chi }$ ${\displaystyle \chi }$
${\displaystyle \mathbb {C} _{\text{class}}}$ ${\displaystyle \chi }$

${\displaystyle \chi _{\mathbb {C} }:R(G)\otimes \mathbb {C} \longrightarrow \mathbb {C} _{\text{class}}(G),}$

by continuing the mapping on a basis of pure tensors defined by or and then bilinear . ${\displaystyle (\tau _{j}\otimes 1)_{j=1,\dotsc ,m}}$ ${\displaystyle \chi _{\mathbb {C} }(\tau _{j}\otimes 1)=\chi _{j}}$ ${\displaystyle \chi _{\mathbb {C} }(\tau _{j}\otimes z)=z\chi _{j},}$

We write for the set of all characters and for the group created by, that is, for the set of all differences between two characters. It applies ${\displaystyle {\mathcal {R}}^{+}(G)}$ ${\displaystyle G}$ ${\displaystyle {\mathcal {R}}(G)}$ ${\displaystyle {\mathcal {R}}^{+}(G)}$

${\displaystyle {\mathcal {R}}(G)=\mathbb {Z} \chi _{1}\oplus \cdots \oplus \mathbb {Z} \chi _{m}\,\,\,{\text{und}}\,\,\,{\mathcal {R}}(G)={\text{Im}}(\chi )=\chi (R(G)).}$

Thus, virtual characters and virtual representations correspond in an optimal way. ${\displaystyle R(G)\cong {\mathcal {R}}(G),}$

There 's the crowd of all of the virtual characters. Since the product of two characters yields a character, a subring of the ring of all class functions is on Since they form a basis of , we get, as we did for isomorphism ${\displaystyle {\text{Im}}(\chi )={\mathcal {R}}(G),}$ ${\displaystyle {\mathcal {R}}(G)}$ ${\displaystyle {\mathcal {R}}(G)}$ ${\displaystyle \mathbb {C} _{\text{class}}(G)}$ ${\displaystyle G.}$ ${\displaystyle \chi _{i}}$ ${\displaystyle \mathbb {C} _{\text{class}}(G)}$ ${\displaystyle R(G),}$ ${\displaystyle \mathbb {C} \otimes {\mathcal {R}}(G)\cong \mathbb {C} _{\text{class}}(G).}$

Restriction and induction

Let the constraint define a ring homomorphism if it is a subgroup of ${\displaystyle H}$ ${\displaystyle G,}$

${\displaystyle {\begin{aligned}{\mathcal {R}}(G)&\to {\mathcal {R}}(H)\\\phi &\mapsto \phi |_{H},\end{aligned}}}$

which we designate with or . The induction on class functions also defines a homomorphism of Abelian groups which is denoted by or . According to Frobenius reciprocity , the two homomorphisms are adjoint to one another with regard to the bilinear forms and the formula also shows ${\displaystyle {\text{Res}}_{H}^{G}}$ ${\displaystyle {\text{Res}}}$ ${\displaystyle \textstyle {\mathcal {R}}(H)\to {\mathcal {R}}(G),}$ ${\displaystyle {\text{Ind}}_{H}^{G}}$ ${\displaystyle {\text{Ind}}}$
${\displaystyle \langle \cdot ,\cdot \rangle _{H}}$ ${\displaystyle \langle \cdot ,\cdot \rangle _{G}.}$

${\displaystyle {\text{Ind}}(\varphi \cdot {\text{Res}}(\psi ))={\text{Ind}}(\varphi )\cdot \psi ,}$

that the image is of an ideal of the ring . Similarly, one can define the mapping via the restriction of representations and the mapping for via induction . With the Frobeniusreziprozität is then obtained that the images are adjoined to each other and that the image of an ideal in is. ${\displaystyle {\text{Ind}}:{\mathcal {R}}(H)\to {\mathcal {R}}(G)}$ ${\displaystyle {\mathcal {R}}(G)}$
${\displaystyle {\text{Res}}}$ ${\displaystyle {\text{Ind}}}$ ${\displaystyle R(G)}$ ${\displaystyle {\text{Im}}({\text{Ind}})={\text{Ind}}(R(H))}$ ${\displaystyle R(G)}$

If there is a commutative ring, the homomorphisms and to -linear mappings can be continued: ${\displaystyle A}$ ${\displaystyle {\text{Res}}}$ ${\displaystyle {\text{Ind}}}$ ${\displaystyle A}$

${\displaystyle {\begin{aligned}A\otimes {\text{Res}}:A\otimes R(G)&\to A\otimes R(H)\\(a\otimes \sum a_{i}\tau _{i})&\mapsto (a\otimes \sum a_{i}{\text{Res}}(\tau _{i}))\end{aligned}}}$

${\displaystyle {\begin{aligned}A\otimes {\text{Ind}}:A\otimes R(H)&\to A\otimes R(G)\\(a\otimes \sum a_{j}\eta _{j})&\mapsto (a\otimes \sum a_{j}{\text{Ind}}(\eta _{j})),\end{aligned}}}$

where the irreducible representations are from up to isomorphism. ${\displaystyle \eta _{j}}$ ${\displaystyle H}$

With we get particular that and homomorphisms between and deliver. ${\displaystyle A=\mathbb {C} }$ ${\displaystyle {\text{Ind}}}$ ${\displaystyle {\text{Res}}}$ ${\displaystyle \mathbb {C} _{\text{class}}(G)}$ ${\displaystyle \mathbb {C} _{\text{class}}(H)}$

Maximum tori

For a compact, connected Lie group one has an isomorphism defined by restriction ${\displaystyle G}$

${\displaystyle R(G)\simeq R(T)^{W}}$,

where is a maximal torus and the acting Weyl group . ${\displaystyle T\subset G}$ ${\displaystyle W}$ ${\displaystyle T}$

Presentation ring of the product of compact groups

All irreducible representations of are exactly those representations for which irreducible representations of or are. This carries over to the presentation ring as identity ${\displaystyle G_{1}\times G_{2}}$ ${\displaystyle \eta _{1}\otimes \eta _{2}}$ ${\displaystyle \eta _{1},\eta _{2}}$ ${\displaystyle G_{1}}$ ${\displaystyle G_{2}}$

${\displaystyle R(G_{1}\times G_{2})=R(G_{1})\otimes _{\mathbb {Z} }R(G_{2}).}$

Artin's theorem

→ Main article : “Artin's theorem” in the article Induced representation

Let be a family of subsets of a finite group Let be the homomorphism defined by the family of Then the following properties are equivalent: ${\displaystyle X}$ ${\displaystyle G.}$ ${\displaystyle {\text{Ind}}\colon \bigoplus _{H\in X}R(H)\to R(G)}$ ${\displaystyle {\text{Ind}}_{H}^{G},\,\,H\in X.}$

• The coke of is finite. ${\displaystyle {\text{Ind}}:\bigoplus _{H\in X}R(H)\to R(G)}$
• ${\displaystyle G}$is the union of the conjugates of the subgroups belonging to it, ie ${\displaystyle X}$ ${\displaystyle G=\bigcup _{H\in X \atop s\in G}sHs^{-1}.}$

Relationship to K-theory

The representation ring is isomorphic to the algebraic K-theory of group algebra:

${\displaystyle R(G)\simeq K_{0}(\mathbb {C} G)}$.

The representation ring of a compact Lie group is isomorphic to the equivariant K-theory of the point:

${\displaystyle R(G)\simeq K_{G}(*)}$.

literature

• Jean-Pierre Serre: Linear Representations of Finite Groups. Springer Verlag, New York 1977.
• 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

Web links

• Representation Ring (nLab)

Individual evidence