nLab vector G-space

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):

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

(Triantafillou 82, (4.1), Golasinski 97a, Lemma 1.1, Scull 08, Def. 2.2, Lemma 2.3)

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):

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

(Triantafillou 82, Section 3 and p. 10, Scull 08, Lemma 2.4, Prop. 2.5)

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).

(Triantafillou 82, Prop. 4.3)

Corollary

Any equivariant PL de Rham complex (Def. ) is a fibrant object in the model structure on equivariant connective dgc-algebras.

(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. .

(Triantafillou 82, p. 10, Scull 01, Prop. 7.34, Scull 08, Def. 2.6)

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:

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

Related concepts

• equivariant chain complex

• equivariant dgc-algebra, model structure on equivariant dgc-algebras

• equivariant rational homotopy theory

References

• Georgia Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) (jstor:1999119)

• Marek Golasiński, Componentwise injective models of functors to DGAs, Colloquium Mathematicum, Vol. 73, No. 1 (1997) (dml:21048, pdf)

• Marek Golasiński, Injective models of $G$-disconnected simplicial sets, Annales de l’Institut Fourier, Volume 47 (1997) no. 5, p. 1491-1522 (numdam:AIF_1997__47_5_1491_0)

• Laura Scull, Rational $S^1$-equivariant homotopy theory, Transactions of the AMS, Volume 354, Number 1, Pages 1-45 2001 (pdf, doi:10.1090/S0002-9947-01-02790-8)

• Laura Scull, A model category structure for equivariant algebraic models, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 (doi:10.1090/S0002-9947-07-04421-2)