Redirected from "dual vector G-space".
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Definition
Let G G be a finite group . We write
(1) G Orbits ⊂ Categories
G Orbits \;\subset\; Categories
for the orbit category of G 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 :
Vector G Spaces ℚ fin ≔ PSh ( G Orbits , VectorSpaces ℚ fin ) ≔ Functors ( G Orbits op , VectorSpaces ℚ 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) ( Vector G Spaces ℚ fin ) op ≃ PSh ( G Orbits op , VectorSpaces ℚ fin ) = Functors ( G Orbits , VectorSpaces ℚ 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) DualVector G Spaces ≔ Functors ( G Orbits , VectorSpaces ℚ )
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
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
(T82, Def. 3.1 )
For
define
V ̲ H ∈ Mod ℚ Orb G op V ̲ H ≡ ℚ [ Orb G ( − , G / H ) ] ⊗ ℚ [ N ( H ) / H ] V H .
\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}
Proposition
(T82, Prop. 3.2 ) The objects V ̲ H \underline{V}_H from Def. are projective in Mod ℚ Orb G op Mod_{\mathbb{Q}}^{Orb_G^{op}} .
Proof
We need to show that dashed lifts in the following diagrams exist, where p 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 Orb_G -component of this diagram separately has such a lift, in particular we may choose a lift f G / H ^ \widehat{f_{G/H}} at stage G / H G/H :
where we have identified V H 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 M ), and it is a lift by naturality of p p and f f :
p G / K ∘ f ^ G / K ( ( G / K → ϕ G / H ) ⊗ v H ) ≡ p G / K ∘ M ( ϕ ) ∘ f G / H ^ ( ( G / H → id G / H ) ⊗ v H ) = N ( ϕ ) ∘ p G / H ∘ f G / H ^ ( ( G / H → id G / H ) ⊗ v H ) = N ( ϕ ) ∘ f G / H ( ( G / H → id G / H ) ⊗ v H ) = f G / K ∘ V ̲ H ( ϕ ) ( ( G / H → id G / H ) ⊗ v H ) = f G / K ( ( G / K → ϕ G / H ) ⊗ v H ) .
\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}
Proof
We make a bunch of choices:
First, in each conjugacy class [ H ] [H] of subgroups G G choose one representative H ⊂ G H \subset G .
For that H ↪ G H \hookrightarrow G , consider the joint span of the images of
P ( G / H ↠ G / H ′ ) : P ( G / H ′ ) → P ( G / H )
P(G/H \twoheadrightarrow G/H')
\;\colon\;
P(G/H') \to P(G/H)
for all intermediate subgroup -inclusions H ↪ H ′ ↪ G H \hookrightarrow H' \hookrightarrow G :
0 → ∑ H ′ ⊃ H im ( P ( G / H ↠ G / H ′ ) ) ↪ P ( G / H ) ↠ P ( G / H ) / ∑ H ′ ⊃ H im ( P ( G / H ↠ G / H ′ ) ) → 0 .
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) σ H : P ( G / H ) / ∑ H ′ ⊃ H im ( P ( G / H ↠ G / H ′ ) ) ↪ − − − P ( G / H ) ,
\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 ≔ σ ( P ( G / H ) / ∑ H ′ ⊃ H im ( P ( G / H ↠ G / H ′ ) ) ) ↪ − − − P ( G / H ) .
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 N(H)/H -equivariantly: Since we are in characteristic zero this follows by the fact that every N ( H ) / H 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 ℚ Orb G op Mod_{\mathbb{Q}}^{Orb_G^{op}} as follows, out of the direct sum of their underlined versions from :
(6) p : ⊕ [ H ] V ̲ H ⟶ P p G / K : ⊕ [ H ] ℚ [ Orb G ( G / K , G / H ) ⊗ ℚ [ N ( H ) / H ] V H ] ⟶ P ( G / K ) ( G / K → f G / H ) ⊗ v H ↦ P ( f ) ( v H ) ,
\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 G/K (via contravariant functoriality of P 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 ⊂ G K \subset G the morphism p G / K p_{G/K} (6) is both an epimorphism and a monomorphism .
First to see that that p G / K p_{G/K} is an epimorphism: To start with, it is clearly surjective onto the summand V K V_K . Hence it is next sufficient to show that given v K ∈ P ( G / K ) v_K \in P(G/K) which is in the image under P ( G / K ↠ G / H ) P(G/K \twoheadrightarrow G/H) of some v ^ H ∈ P ( G / H ) \widehat{v}_H \in P(G/H) then it is also in the image of p G / K p_{G/K} . As before, this is clear for those v ^ H ∈ V H \widehat{v}_H \in V_H . Hence next, as before, it is sufficient to show this for those v ^ H \widehat{v}_H which are in the image under some P ( G / H ↠ G / H ′ ) P(G/H \twoheadrightarrow G/H') of some v H ′ ∈ P ( G / H ′ ) v_{H'} \in P(G/H') … And so on. Since G G is a finite group , this recursive argument eventually terminates with V G = P ( G / G ) V_G = P(G/G) .
Finally, to see that p G / K p_{G/K} (6) is a monomorphism. It is here (only) that we use the assumption that P P is projective . With the previous point, this implies a lift p ′ p' in the following diagram in Mod ℚ Orb G op Mod_{\mathbb{Q}}^{Orb_G^{op}} :
Hence if v , w ∈ P ( G / K ) v,w \,\in\, P(G/K) such that p ′ G / K ( v ) = p ′ G / K ( w ) p'_{G/K}(v) = p'_{G/K}(w) then p G / K ∘ p ′ G / K ( v ) = p G / k ∘ p ′ G / K ( w ) p_{G/K} \circ p'_{G/K}(v) = p_{G/k} \circ p'_{G/K}(w) hence v = w v = w , whence each p ′ G / K p'_{G/K} is injective.
Corollary
(T82, Prop. 3.6 ) Every object N ∈ Mod ℚ Orb G N \,\in\, Mod_{\mathbb{Q}}^{Orb_G} admits a projective cover in the sense of a projective object ⊕ [ H ] V ̲ H \underset{[H]}{\oplus} \underline{V}_H and an epimorphism p : ⊕ [ H ] V ̲ H ↠ N 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
define
V ̲ H ∈ Mod ℚ Orb G V ̲ H ≡ Mod ℚ [ N ( H ) / H ] ( ℚ [ Orb G ( − , G / H ) ] , V H ) .
\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}
Proof
We make a bunch of choices:
First, in each conjugacy class [ H ] [H] of subgroups G G choose one representative H ⊂ G H \subset G .
For that H ↪ G H \hookrightarrow G , consider the intersection of the kernels of
I ( G / H ↠ G / H ′ ) : I ( G / H ) → I ( G / H ′ )
I(G/H \twoheadrightarrow G/H')
\;\colon\;
I(G/H) \to I(G/H')
for all intermediate subgroup -inclusions H ↪ H ′ ↪ G H \hookrightarrow H' \hookrightarrow G ,
and an N ( H ) / H N(H)/H -equivariant splitting
(7) τ H : I ( G / H ) ↠ ⋂ H ′ ⊃ H ker ( I ( G / H ↠ G / H ′ ) ) ≡ V H .
\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 ℚ Orb G Mod_{\mathbb{Q}}^{Orb_G} as follows,
(8) i : I ⟶ ⊕ [ H ] V ̲ H i G / K : I ( G / K ) ⟶ ⊕ [ H ] Mod ℚ [ N ( H ) / H ] ( ℚ [ Orb G ( G / K , G / H ) , V H ] ) i K ↦ ⊕ [ H ] ( ( G / K → f G / H ) ↦ τ H ∘ I ( f ) ( i K ) ) ,
\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 I which implies both that the maps on the right are N ( H ) / H N(H)/H -equivariant and that this transformation is natural in G / K G/K :
Now to check that this map i i is in fact an isomorphism if I 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 G -spaces to Weyl group representations)
Let H ⊂ G 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 ( G / H , G / H ) ≃ Aut G Orbits ( 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 a full subcategory -inclusion
B W G ( H ) ↪ i H G Orbits
\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 G (1) , and hence a restriction functor
(10) W G ( H ) Representations l , ℚ fin ⟵ ( − ) * ∘ i H * ( Vector G Spaces ℚ 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
(11) W G ( H ) Representations r , ℚ fin ⟵ i H * DualVector G Spaces
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 G -spaces)
For H ⊂ G H \subset G a subgroup and
V ∈ W G ( H ) Representations l , ℚ fin
V \;\in\; W_G(H) Representations^{fin}_{l,\mathbb{Q}}
a rational finite dimensional left representation of the Weyl group of H H in G G , write
G Orbits ⟶ I H ( V ) ℚ VectorSpaces G / K ↦ W G ( H ) Representations ( ℚ [ G Orbits ( 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 G G -orbit category to rational vector spaces which assigns to a coset space G / K G/K the vector space of homomorphisms of right actions by the Weyl group (9) from the hom-set G Orbits ( 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 , ℚ
V^\ast \;\in\; W_G(H) Representations_{r,\mathbb{Q}}
set
G Orbits ⟶ I H ( V * ) ℚ VectorSpaces G / K ↦ W G ( H ) Representations ( ℚ [ G Orbits ( 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 (10) :
W G ( H ) Representations ℚ ⊥ ⟶ I H ⟵ ( − ) * ∘ i H * DualVector G Spaces
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 G G -space is a direct sum of objects of this form, specifically (see Def. below):
V ̲ ∈ DualVector G Spaces is injective ⇔ V ̲ ≃ ⨁ [ H ⫋ G ] conj I H ( ⋂ K ⊃ H ker ( V ̲ ( G / H ) ⟶ V ̲ ( G / H → G / K ) V ̲ ( 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 G G -space)
Let S ∈ G SimplicialSets S \in G SimplicialSets a simplicial set equipped with G G -action , say that the equivariant PL de Rham complex is the functor on the orbit category
G Orbits ⟶ 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 / H G/H assigns the PL de Rham complex of the H H -fixed locus X H ⊂ X X^H \subset X .
Then the underlying dual vector G-space
G Orbits ⟶ Ω PLdR • ( Maps ( − , X ) G ) dgcAlgebras ⟶ VectorSpaces ℚ
\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 V ̲ ∈ DualVector G Spaces \underline{V} \in DualVector G Spaces (3) , its injective envelope is
⨁ [ H ⊂ G ] conj I H ( ⋂ K ⫌ H ker ( V ̲ ( G / H ) ⟶ V ̲ ( G / H → G / K ) V ̲ ( 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
the direct sum is over conjugacy classes of subgroups , with H ⊂ G H \subset G on the right any one representative of its conjugacy class,
for H = G H = G the argument of I H I_H is taken to be all of V ̲ ( G / G ) \underline{V}(G/G) ,
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 )
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 = ℤ 2 G = \mathbb{Z}_2
Example
(orbit category of Z/2Z )
For equivariance group the cyclic group of order 2 :
G ≔ ℤ 2 ≔ ℤ / 2 ℤ .
G \;\coloneqq\; \mathbb{Z}_2 \;\coloneqq\; \mathbb{Z}/2\mathbb{Z}
\,.
the orbit category looks like this:
(12) ℤ 2 Orbits = { ℤ 2 / 1 ⟶ AAAAA ℤ 2 / ℤ 2 Aut = ℤ 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.:
ℤ 2 Orbits ( ℤ 2 / ℤ 2 , ℤ 2 / ℤ 2 ) ≃ 1 ℤ 2 Orbits ( ℤ 2 / 1 , ℤ 2 / ℤ 2 ) ≃ * ℤ 2 Orbits ( ℤ 2 / ℤ 2 , ℤ 2 / 1 ) ≃ ∅ ℤ 2 Orbits ( ℤ 2 / 1 , ℤ 2 / 1 ) ≃ ℤ 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 ∈ ℤ 2 Representations
\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 ℤ 2 ( 1 ) = ℤ 2 W_{\mathbb{Z}_2}(1) = \mathbb{Z}_2 .
Their induced injective dual vector ℤ 2 \mathbb{Z}_2 -spaces, according to Def. , are:
I 1 ( 1 ) : ℤ 2 / 1 ↦ ℤ 2 Reps ( ℚ [ ℤ 2 Orbits ( ℤ 2 / 1 , ℤ 2 / 1 ) ] ⏟ ≃ 1 ⊕ 1 sgn , 1 ) ≃ 1 ↓ ℤ 2 / ℤ 2 ↦ ℤ 2 Reps ( ℚ [ ℤ 2 Orbits ( ℤ 2 / ℤ 2 , ℤ 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 ) : ℤ 2 / 1 ↦ ℤ 2 Reps ( ℚ [ ℤ 2 Orbits ( ℤ 2 / 1 , ℤ 2 / 1 ) ] ⏟ ≃ 1 ⊕ 1 sgn , 1 sgn ) ≃ 1 sgn ↓ ℤ 2 / ℤ 2 ↦ ℤ 2 Reps ( ℚ [ ℤ 2 Orbits ( ℤ 2 / ℤ 2 , ℤ 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
1 ∈ 1 Representations
\mathbf{1} \;\in\; 1 Representations
for the unique irrep of the Weyl group W ℤ 2 ( ℤ 2 ) = 1 W_{\mathbb{Z}_2}(\mathbb{Z}_2) = 1 .
Its induced injective dual vector ℤ 2 \mathbb{Z}_2 -spaces, according to Def. , is:
I ℤ 2 ( 1 ) : ℤ 2 / 1 ↦ 1 Reps ( ℚ [ ℤ 2 Orbits ( ℤ 2 / 1 , ℤ 2 / ℤ 2 ) ] ⏟ ≃ 1 , 1 ) ≃ 1 ↓ ↓ id ℤ 2 / ℤ 2 ↦ 1 Reps ( ℚ [ ℤ 2 Orbits ( ℤ 2 / ℤ 2 , ℤ 2 / ℤ 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
Georgia Triantafillou , Equivariant minimal models , Trans. Amer. Math. Soc. 274 (1982) 509-532 [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 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 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 )