# nLab vector G-space

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

and

# Contents

## Definition

Let $G$ be a finite group. We write

(1)$G Orbits \;\subset\; Categories$

for the orbit category of $G$.

###### Definition

(rational vector G-spaces)

We say that the category of finite-dimensional vector G-spaces is the category of functors from the opposite of the orbit category to the category of finite-dimensional rational vector spaces:

\begin{aligned} Vector G Spaces^{fin}_{\mathbb{Q}} & \coloneqq \; PSh \Big( G Orbits \,,\, VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}} \Big) \\ & \coloneqq \; Functors \Big( G Orbits^{op} \,,\, VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}} \Big) \end{aligned}

Its opposite category we call the category of finite-dimensional dual vector G-spaces:

(2)\begin{aligned} \big( Vector G Spaces^{fin}_{\mathbb{Q}} \big)^{op} & \simeq \; PSh \Big( G Orbits^{op} \,,\, VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}} \Big) \\ & = \; Functors \Big( G Orbits \,,\, VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}} \Big) \end{aligned}

(using in the first line that forming dual linear maps is an equivalence of categories from finite dimensional vector spaces to their opposite category.)

In generalization of (2), dropping the finiteness condition, we write

(3)$DualVector G Spaces \;\coloneqq\; Functors \Big( G Orbits \,,\, VectorSpaces_{\mathbb{Q}} \Big)$

The category (3) is denoted $Vec_G^\ast$ in (Triantafillou 82).

## Properties

### Injective objects

###### Example

(restriction of vector $G$-spaces to Weyl group representations)

Let $H \subset G$ any subgroup. Notice that its Weyl group is the automorphism group of its coset space in the orbit category:

(4)$G Orbits \big( G/H \,,\, G/H \big) \;\; \simeq \;\; Aut_{G Orbits} \big( G/H \big) \;\; \simeq \;\; W_G(H) \;\; \coloneqq \;\; N_G(H)/H$

This gives an full subcategory-inclusion

$\mathbf{B} W_G(H) \; \overset{\;\;i_H\;\;}{\hookrightarrow} \; G Orbits$

of the delooping category of they Weyl group into the orbit category of $G$ (1), and hence a restriction functor

(5)$W_G(H) Representations^{fin}_{l,\mathbb{Q}} \overset{ \;\;\; (-)^\ast \,\circ\, i_H^\ast \;\;\; }{\longleftarrow} \big( Vector G Spaces_{\mathbb{Q}}^{fin} \big)^{op}$

or more generally

(6)$W_G(H) Representations^{fin}_{r,\mathbb{Q}} \overset{ \;\;\; i_H^\ast \;\;\; }{\longleftarrow} DualVector G Spaces$

By the general end-formula for right Kan extension (here), this restriction functor has a right adjoint, given as follows:

###### Definition

(injective atoms of dual vector $G$-spaces)

For $H \subset G$ a subgroup and

$V \;\in\; W_G(H) Representations^{fin}_{l,\mathbb{Q}}$

a rational finite dimensional left representation of the Weyl group of $H$ in $G$, write

$\array{ G Orbits & \overset{ I_H(V) }{\longrightarrow} & \mathbb{Q}VectorSpaces \\ G/K &\mapsto& W_G(H) Representations \Big( \mathbb{Q} \big[ G Orbits ( G/K, G/H ) \big] \,,\, V^\ast \Big) }$

for the functor from the $G$-orbit category to rational vector spaces which assigns to a coset space $G/K$ the vector space of homomorphisms of right actions by the Weyl group (4) from the hom-set $G Orbits\big(G/K, G/H \big)$ to the dual vector space equipped with its dual action.

More generally, for

$V^\ast \;\in\; W_G(H) Representations_{r,\mathbb{Q}}$

set

$\array{ G Orbits & \overset{ I_H(V^\ast) }{\longrightarrow} & \mathbb{Q}VectorSpaces \\ G/K &\mapsto& W_G(H) Representations \Big( \mathbb{Q} \big[ G Orbits ( G/K, G/H ) \big] \,,\, V^\ast \Big) }$

This construction extends to a functor right adjoint to the restriction (5):

$W_G(H) Representations_{\mathbb{Q}} \underoverset{ \underset{ \;\;\; I_H \;\;\; }{ \longrightarrow } }{ \overset{ \;\;\; (-)^\ast \,\circ\, i_H^\ast \;\;\; }{\longleftarrow} } {\bot} DualVector G Spaces$
###### Proposition

$\,$

(i) The objects of the form $I_H(V^\ast)$ (Def. ) are injective objects in dual vector G-spaces (Def. ).

(ii) Every injective dual vector $G$-space is a direct sum of objects of this form, specifically (see Def. below):

$\underline{V} \;\in\; DualVector G Spaces \;\;\; \text{is injective} \;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\; \underline{V} \;\simeq\; \underset{ \big[ H \subsetneqq G \big]_{conj} }{\bigoplus} \, I_H \left( \underset{ K \supset H }{\bigcap} ker \big( \underline{V}(G/H) \overset{ \underline{V}(G/H \to G/K) }{\longrightarrow} \underline{V}(G/K) \big) \right)$
###### Example

(equivariant PL de Rham complex in injective dual vector $G$-space)

Let $S \in G SimplicialSets$ a simplicial set equipped with $G$-action, say that the equivariant PL de Rham complex is the functor on the orbit category

$\array{ G Orbits & \overset{ \Omega^\bullet_{PLdR} \big( Maps(-,X)^G \big) }{\longrightarrow} & dgcAlgebras \\ G/H &\mapsto& \Omega^\bullet_{PLdR} \big( X^H \big) }$

which to a coset space $G/H$ assigns the PL de Rham complex of the $H$-fixed locus $X^H \subset X$.

Then the underlying dual vector G-space

$\array{ G Orbits & \overset{ \Omega^\bullet_{PLdR} \big( Maps(-,X)^G \big) }{\longrightarrow} & dgcAlgebras &\overset{}{\longrightarrow}& VectorSpaces_{\mathbb{Q}} }$

is an injective object (degreewise, in fact).

###### Corollary

(also Scull 08, Lemma 5.2)

###### Definition

(injective envelope of dual vector G-spaces)

For $\underline{V} \in DualVector G Spaces$ (3), its injective envelope is

$\underset{ \big[ H \subset G \big]_{conj} }{\bigoplus} \, I_H \left( \underset{ K \supsetneqq H }{\bigcap} ker \big( \underline{V}(G/H) \overset{ \underline{V}(G/H \to G/K) }{\longrightarrow} \underline{V}(G/K) \big) \right) \,,$

where

1. the direct sum is over conjugacy classes of subgroups, with $H \subset G$ on the right any one representative of its conjugacy class,

2. for $H = G$ the argument of $I_H$ is taken to be all of $\underline{V}(G/G)$,

3. $I_H(-)$ is the injective atom construction from Def. .

###### Lemma

(tensor product of injective dual vector $G$-spaces)

The object-wise tensor product of two finite-dimensional injective dual vector G-spaces (Def. ) is again injective.

This is proven as Golasinski 97b, Lemma 3.6 (use Golasinski 97b, Remark 1.2 to see that the Lemma does apply to the ordinary tensor product of finite-dimensional vector spaces).

Beware that incorrect versions of this statement had been circulating; for discussion of the literature see Golasinski 97b, p. 3 and Scull 01, Prop. 7.36

## Examples

### Over $G = \mathbb{Z}_2$

###### Example

(orbit category of Z/2Z)

For equivariance group the cyclic group of order 2:

$G \;\coloneqq\; \mathbb{Z}_2 \;\coloneqq\; \mathbb{Z}/2\mathbb{Z} \,.$

the orbit category looks like this:

(7)$\mathbb{Z}_2 Orbits \;=\; \left\{ \array{ \mathbb{Z}_2/1 & \overset{ \phantom{AAAAA} }{ \longrightarrow } & \mathbb{Z}_2/\mathbb{Z}_2 \\ Aut = \mathbb{Z}_2 && Aut = 1 } \right\}$

i.e.:

\begin{aligned} \mathbb{Z}_2 Orbits \big( \mathbb{Z}_2/\mathbb{Z}_2 \,,\, \mathbb{Z}_2/\mathbb{Z}_2 \big) \;\simeq\; 1 \\ \mathbb{Z}_2 Orbits \big( \mathbb{Z}_2/1 \,,\, \mathbb{Z}_2/\mathbb{Z}_2 \big) \;\simeq\; \ast \\ \mathbb{Z}_2 Orbits \big( \mathbb{Z}_2/\mathbb{Z}_2 \,,\, \mathbb{Z}_2/1 \big) \;\simeq\; \varnothing \\ \mathbb{Z}_2 Orbits \big( \mathbb{Z}_2/1 \,,\, \mathbb{Z}_2/1 \big) \;\simeq\; \mathbb{Z}_2 \end{aligned}

Write

$\mathbf{1}, \mathbf{1}_{sgn} \;\in\; \mathbb{Z}_2 Representations$

for the two irreducible representations (the trivial representation and the sign representation, respectively) of the Weyl group $W_{\mathbb{Z}_2}(1) = \mathbb{Z}_2$.

Their induced injective dual vector $\mathbb{Z}_2$-spaces, according to Def. , are:

$I_1(\mathbf{1}) \;\; \colon \;\; \;\;\;\;\; \array{ \mathbb{Z}_2/1 &\mapsto& \mathbb{Z}_2 Reps \Big( \underset{ \simeq \, \mathbf{1} \oplus \mathbf{1}_{sgn} }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/1 ) \big] } } \,,\, \mathbf{1} \Big) & \simeq & \mathbf{1} \\ \big\downarrow && \\ \mathbb{Z}_2/\mathbb{Z}_2 &\mapsto& \mathbb{Z}_2 Reps \Big( \underset{ \simeq \, 0 }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/1 ) \big] } } \,,\, \mathbf{1} \Big) & \simeq & 0 }$

and

$I_1(\mathbf{1}_{sgn}) \;\; \colon \;\; \;\;\;\;\; \array{ \mathbb{Z}_2/1 &\mapsto& \mathbb{Z}_2 Reps \Big( \underset{ \simeq \, \mathbf{1} \oplus \mathbf{1}_{sgn} }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/1 ) \big] } } \,,\, \mathbf{1}_{sgn} \Big) & \simeq & \mathbf{1}_{sgn} \\ \big\downarrow && \\ \mathbb{Z}_2/\mathbb{Z}_2 &\mapsto& \mathbb{Z}_2 Reps \Big( \underset{ \simeq \, 0 }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/1 ) \big] } } \,,\, \mathbf{1}_{sgn} \Big) & \simeq & 0 }$

Similarly, write

$\mathbf{1} \;\in\; 1 Representations$

for the unique irrep of the Weyl group $W_{\mathbb{Z}_2}(\mathbb{Z}_2) = 1$.

Its induced injective dual vector $\mathbb{Z}_2$-spaces, according to Def. , is:

$I_{\mathbb{Z}_2}(\mathbf{1}) \;\; \colon \;\; \;\;\;\;\; \array{ \mathbb{Z}_2/1 &\mapsto& 1 Reps \Big( \underset{ \simeq \, \mathbf{1} }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/\mathbb{Z}_2 ) \big] } } \,,\, \mathbf{1} \Big) & \simeq & \mathbf{1} \\ \big\downarrow && && \big\downarrow{}^{\mathrlap{\mathrm{id}}} \\ \mathbb{Z}_2/\mathbb{Z}_2 &\mapsto& 1 Reps \Big( \underset{ \simeq \, \mathbf{1} }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/\mathbb{Z}_2 ) \big] } } \,,\, \mathbf{1} \Big) & \simeq & \mathbf{1} }$