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vector G-space

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Representation theory

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

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Rational homotopy theory

Contents

Definition

Let GG be a finite group. We write

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

for the orbit category of GG.

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:

VectorGSpaces fin PSh(GOrbits,VectorSpaces fin) Functors(GOrbits op,VectorSpaces fin) \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)(VectorGSpaces fin) op PSh(GOrbits op,VectorSpaces fin) =Functors(GOrbits,VectorSpaces fin) \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)DualVectorGSpacesFunctors(GOrbits,VectorSpaces ) DualVector G Spaces \;\coloneqq\; Functors \Big( G Orbits \,,\, VectorSpaces_{\mathbb{Q}} \Big)

The category (3) is denoted Vec G *Vec_G^\ast in (Triantafillou 82).

Properties

Injective objects

Example

(restriction of vector GG-spaces to Weyl group representations)

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

(4)GOrbits(G/H,G/H)Aut GOrbits(G/H)W G(H)N G(H)/H 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

BW G(H)i HGOrbits \mathbf{B} W_G(H) \; \overset{\;\;i_H\;\;}{\hookrightarrow} \; G Orbits

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

(5)W G(H)Representations l, fin() *i H *(VectorGSpaces fin) op 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 r, fini H *DualVectorGSpaces 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 GG-spaces)

For HGH \subset G a subgroup and

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

a rational finite dimensional left representation of the Weyl group of HH in GG, write

GOrbits I H(V) VectorSpaces G/K W G(H)Representations([GOrbits(G/K,G/H)],V *) \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 GG-orbit category to rational vector spaces which assigns to a coset space G/KG/K the vector space of homomorphisms of right actions by the Weyl group (4) from the hom-set GOrbits(G/K,G/H)G Orbits\big(G/K, G/H \big) to the dual vector space equipped with its dual action.

More generally, for

V *W G(H)Representations r, V^\ast \;\in\; W_G(H) Representations_{r,\mathbb{Q}}

set

GOrbits I H(V *) VectorSpaces G/K W G(H)Representations([GOrbits(G/K,G/H)],V *) \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 I H() *i H *DualVectorGSpaces 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 *)I_H(V^\ast) (Def. ) are injective objects in dual vector G-spaces (Def. ).

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

V̲DualVectorGSpacesis injectiveV̲[HG] conjI H(KHker(V̲(G/H)V̲(G/HG/K)V̲(G/K))) \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 GG-space)

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

GOrbits Ω PLdR (Maps(,X) G) dgcAlgebras G/H Ω PLdR (X H) \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/HG/H assigns the PL de Rham complex of the HH-fixed locus X HXX^H \subset X.

Then the underlying dual vector G-space

GOrbits Ω PLdR (Maps(,X) G) dgcAlgebras VectorSpaces \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)

(also Scull 08, Lemma 5.2)

Definition

(injective envelope of dual vector G-spaces)

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

[HG] conjI H(KHker(V̲(G/H)V̲(G/HG/K)V̲(G/K))), \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 HGH \subset G on the right any one representative of its conjugacy class,

  2. for H=GH = G the argument of I HI_H is taken to be all of V̲(G/G)\underline{V}(G/G),

  3. I H()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 GG-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= 2G = \mathbb{Z}_2

Example

(orbit category of Z/2Z)

For equivariance group the cyclic group of order 2:

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

the orbit category looks like this:

(7) 2Orbits={ 2/1 AAAAA 2/ 2 Aut= 2 Aut=1} \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.:

2Orbits( 2/ 2, 2/ 2)1 2Orbits( 2/1, 2/ 2)* 2Orbits( 2/ 2, 2/1) 2Orbits( 2/1, 2/1) 2 \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

1,1 sgn 2Representations \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 2(1)= 2W_{\mathbb{Z}_2}(1) = \mathbb{Z}_2.

Their induced injective dual vector 2\mathbb{Z}_2-spaces, according to Def. , are:

I 1(1): 2/1 2Reps([ 2Orbits( 2/1, 2/1)]11 sgn,1) 1 2/ 2 2Reps([ 2Orbits( 2/ 2, 2/1)]0,1) 0 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(1 sgn): 2/1 2Reps([ 2Orbits( 2/1, 2/1)]11 sgn,1 sgn) 1 sgn 2/ 2 2Reps([ 2Orbits( 2/ 2, 2/1)]0,1 sgn) 0 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

11Representations \mathbf{1} \;\in\; 1 Representations

for the unique irrep of the Weyl group W 2( 2)=1W_{\mathbb{Z}_2}(\mathbb{Z}_2) = 1.

Its induced injective dual vector 2\mathbb{Z}_2-spaces, according to Def. , is:

I 2(1): 2/1 1Reps([ 2Orbits( 2/1, 2/ 2)]1,1) 1 id 2/ 2 1Reps([ 2Orbits( 2/ 2, 2/ 2)]1,1) 1 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} }

References

Last revised on September 29, 2020 at 06:36:35. See the history of this page for a list of all contributions to it.