# 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

### Projective objects

Beware that this section uses different notational conventions than the rest of the entry. None of the rest of the entry is necessary for reading this section here.

Notation:

###### Definition

For

• $H \subset G$,

• $V_H \,\in\, Mod_{ \mathbb{Q}[N(H)/H] }$

define

$\begin{array}{l} \underline{V}_H \;\in\; Mod_{\mathbb{Q}}^{Orb_G^{op}} \\ \underline{V}_H \;\equiv\; \mathbb{Q}\big[ Orb_G(-,G/H) \big] \underset{ \mathbb{Q}[N(H)/H] }{\otimes} V_H \mathrlap{\,.} \end{array}$

###### Remark

Since the Weyl group of $H$ is (see there) the endomorphism monoid of $G/H$ in the $G$-orbit category

$N(H)/H \;\simeq\; {Orb_G}(G/H,G/H) \;\;$

we have with Def. canonical isomorphism:

$\begin{array}{ccl} \underline{V}_H(G/H) &\equiv& \mathbb{Q}\big[ Orb_G(G/H,G/H) \big] \underset{ \mathbb{Q}[N(H)/H] }{\otimes} V_H \\ &\simeq& V_H \mathrlap{\,.} \end{array}$

###### Proposition

(T82, Prop. 3.2)
The objects $\underline{V}_H$ from Def. are projective in $Mod_{\mathbb{Q}}^{Orb_G^{op}}$.

###### Proof

We need to show that dashed lifts in the following diagrams exist, where $p$ is an epimorphism: Since plain rational vector spaces are free modules (by the basis theorem), hence projective modules, hence projective objects in Vect, each $Orb_G$-component of this diagram separately has such a lift, in particular we may choose a lift $\widehat{f_{G/H}}$ at stage $G/H$:

where we have identified $V_H$ in the bottom left via Rem. .

With this local lift in hand, we obtain a global lift by setting:

This is clearly a natural transformation (by the contravariant functoriality) of $M$), and it is a lift by naturality of $p$ and $f$:

$\begin{array}{l} p_{G/K} \circ \widehat{f}_{G/K} \big( (G/K \xrightarrow{\phi} G/H) \otimes v_H \big) \\ \;\equiv\; p_{G/K} \circ M(\phi) \circ \widehat{f_{G/H}} \big( (G/H \xrightarrow{id} G/H) \otimes v_H \big) \\ \;=\; N(\phi) \circ p_{G/H} \circ \widehat{f_{G/H}} \big( (G/H \xrightarrow{id} G/H) \otimes v_H \big) \\ \;=\; N(\phi) \circ f_{G/H} \big( (G/H \xrightarrow{id} G/H) \otimes v_H \big) \\ \;=\; f_{G/K} \circ \underline{V}_H(\phi) \big( (G/H \xrightarrow{id} G/H) \otimes v_H \big) \\ \;=\; f_{G/K} \big( (G/K \xrightarrow{\phi} G/H) \otimes v_H \big) \mathrlap{\,.} \end{array}$

###### Proposition

(T82, Prop. 3.4)
Every projective object $P \,\in\, Mod_{\mathbb{Q}}^{Orb_G^{op}}$ is a direct sum of projective generators as in Prop. .

###### Proof

We make a bunch of choices:

First, in each conjugacy class $[H]$ of subgroups $G$ choose one representative $H \subset G$.

For that $H \hookrightarrow G$, consider the joint span of the images of

$P(G/H \twoheadrightarrow G/H') \;\colon\; P(G/H') \to P(G/H)$

for all intermediate subgroup-inclusions $H \hookrightarrow H' \hookrightarrow G$:

$0 \to \underset{ H' \supset H }{ \textstyle{\sum} } im\big( P(G/H \twoheadrightarrow G/H') \big) \hookrightarrow P(G/H) \twoheadrightarrow P(G/H) \big/ \underset{ H' \supset H }{ \textstyle{\sum} } im\big( P(G/H \twoheadrightarrow G/H') \big) \to 0 \,.$

On the right we have exhibited the quotient vector space of the inclusion of the joint images on the left (hence the joint cokernel) making a short exact sequence of rational vector spaces.

Since every short exact sequence of vector spaces splits, we may next choose a splitting:

(4)$\sigma_H \;\;\colon\;\; P(G/H) \big/ \underset{ H' \supset H }{ \textstyle{\sum} } im\big( P(G/H \twoheadrightarrow G/H') \big) \xhookrightarrow{\phantom{---}} P(G/H) \,,$

whose image we denote by

(5)$V_H \;\;\;\coloneqq\;\;\; \sigma \Big( P(G/H) \big/ \underset{ H' \supset H }{ \textstyle{\sum} } im\big( P(G/H \twoheadrightarrow G/H') \big) \Big) \xhookrightarrow{\phantom{---}} P(G/H) \,.$

(In fact, we need this splitting $N(H)/H$-equivariantly: Since we are in characteristic zero this follows by the fact that every $N(H)/H$-representation splits as a direct sum of irreducible representations, and by the first part of Schur's lemma, which says that there are no non-zero maps between distinct such direct summands.)

Via these (images of) chosen splittings (5), we may define a morphism in $Mod_{\mathbb{Q}}^{Orb_G^{op}}$ as follows, out of the direct sum of their underlined versions from :

(6)$\array{ p &\colon& \underset{ [H] }{\oplus} \underline{V}_H &\longrightarrow& P \\ p_{G/K} &\colon& \underset{ [H] }{\oplus} \mathbb{Q} \big[ Orb_G(G/K, G/H) \underset{ \mathbb{Q}[N(H)/H] }{\otimes} V_H \big] &\longrightarrow& P(G/K) \\ && \big( G/K \xrightarrow{ f } G/H \big) \,\otimes\, v_H &\mapsto& P(f)(v_H) \,, }$

which is manifestly natural in $G/K$ (via contravariant functoriality of $P$) and hence well-defined:

Since all the direct summands on the left are projective by Prop. , it is now sufficient to prove that (6) is an isomorphism. Since isomorphisms in functor categories are detected objectwise and since rational vector spaces form a balanced category (see there) for this it is sufficient to show that for all $K \subset G$ the morphism $p_{G/K}$ (6) is both an epimorphism and a monomorphism.

First to see that that $p_{G/K}$ is an epimorphism: To start with, it is clearly surjective onto the summand $V_K$. Hence it is next sufficient to show that given $v_K \in P(G/K)$ which is in the image under $P(G/K \twoheadrightarrow G/H)$ of some $\widehat{v}_H \in P(G/H)$ then it is also in the image of $p_{G/K}$. As before, this is clear for those $\widehat{v}_H \in V_H$. Hence next, as before, it is sufficient to show this for those $\widehat{v}_H$ which are in the image under some $P(G/H \twoheadrightarrow G/H')$ of some $v_{H'} \in P(G/H')$… And so on. Since $G$ is a finite group, this recursive argument eventually terminates with $V_G = P(G/G)$.

Finally, to see that $p_{G/K}$ (6) is a monomorphism. It is here (only) that we use the assumption that $P$ is projective. With the previous point, this implies a lift $p'$ in the following diagram in $Mod_{\mathbb{Q}}^{Orb_G^{op}}$:

Hence if $v,w \,\in\, P(G/K)$ such that $p'_{G/K}(v) = p'_{G/K}(w)$ then $p_{G/K} \circ p'_{G/K}(v) = p_{G/k} \circ p'_{G/K}(w)$ hence $v = w$, whence each $p'_{G/K}$ is injective.

###### Corollary

(T82, Prop. 3.6) Every object $N \,\in\, Mod_{\mathbb{Q}}^{Orb_G}$ admits a projective cover in the sense of a projective object $\underset{[H]}{\oplus} \underline{V}_H$ and an epimorphism $p \,\colon\,\underset{[H]}{\oplus} \underline{V}_H \twoheadrightarrow N$.

###### Proof

The construction and verification is verbatim as in the proof of Prop. , omitting only the proof of injectivity in the last step.

### Injective objects (1)

We spell out aspects of the discussion of injective objects in the copresheaf category dual to the above discussion of projective objects in the presheaf category but left implicit in Triantafillou 1982, p. 517.

Notation:

###### Definition

(dual to T82, Def. 3.1)

For

• $H \subset G$,

• $V^H \,\in\, Mod_{ \mathbb{Q}[N(H)/H] }$

define

$\begin{array}{l} \underline{V}^H \;\in\; Mod_{\mathbb{Q}}^{Orb_G} \\ \underline{V}^H \;\equiv\; Mod_{\mathbb{Q}[N(H)/H]} \Big( \mathbb{Q}\big[ Orb_G(-,G/H) \big] ,\, V^H \Big) \mathrlap{\,.} \end{array}$

###### Proposition

(dual to T82, Prop. 3.4)
Every injective object $I \,\in\, Mod_{\mathbb{Q}}^{Orb_G}$ is a direct sum of injective generators as in Prop. .

###### Proof

We make a bunch of choices:

First, in each conjugacy class $[H]$ of subgroups $G$ choose one representative $H \subset G$.

For that $H \hookrightarrow G$, consider the intersection of the kernels of

$I(G/H \twoheadrightarrow G/H') \;\colon\; I(G/H) \to I(G/H')$

for all intermediate subgroup-inclusions $H \hookrightarrow H' \hookrightarrow G$,

and an $N(H)/H$-equivariant splitting

(7)$\tau_H \;\;\colon\;\; I(G/H) \twoheadrightarrow \underset{ H' \supset H }{ \textstyle{\bigcap} } ker\big( I(G/H \twoheadrightarrow G/H') \big) \,\equiv\, V^H \,.$

With this, we may define a morphism in $Mod_{\mathbb{Q}}^{Orb_G}$ as follows,

(8)$\array{ i &\colon& I &\longrightarrow& \underset{ [H] }{\oplus} \underline{V}^H \\ i_{G/K} &\colon& I(G/K) &\longrightarrow& \underset{ [H] }{\oplus} Mod_{\mathbb{Q}[N(H)/H]} \Big( \mathbb{Q} \big[ Orb_G(G/K, G/H) ,\, V^H \big] \Big) \\ && i_K &\mapsto& \underset{[H]}{\oplus} \Big( \big( G/K \xrightarrow{f} G/H \big) \mapsto \tau_{H} \circ I(f)(i_K) \Big) \,, }$

where it is the functoriality of $I$ which implies both that the maps on the right are $N(H)/H$-equivariant and that this transformation is natural in $G/K$:

Now to check that this map $i$ is in fact an isomorphism if $I$ is injective (…)

(…)

### Injective objects (2)

The following is another survey of aspects of the injective objects from Triantafillou 1982. This is from a different edit using different notational conventions than the previous subsection. Eventually both subsections should be harmonized and merged.

###### 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:

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

This gives a 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

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

or more generally

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

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 (9) 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 (10):

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

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

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:

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