Contents
Context
Representation theory
representation theory
geometric representation theory
Ingredients
Definitions
representation, 2-representation, ∞-representation
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group, ∞-group
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group algebra, algebraic group, Lie algebra
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vector space, n-vector space
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affine space, symplectic vector space
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action, ∞-action
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module, equivariant object
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bimodule, Morita equivalence
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induced representation, Frobenius reciprocity
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Hilbert space, Banach space, Fourier transform, functional analysis
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orbit, coadjoint orbit, Killing form
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unitary representation
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geometric quantization, coherent state
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socle, quiver
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module algebra, comodule algebra, Hopf action, measuring
Geometric representation theory
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D-module, perverse sheaf,
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Grothendieck group, lambda-ring, symmetric function, formal group
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principal bundle, torsor, vector bundle, Atiyah Lie algebroid
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geometric function theory, groupoidification
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Eilenberg-Moore category, algebra over an operad, actegory, crossed module
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reconstruction theorems
Theorems
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
Theorems
Rational homotopy theory
Contents
Definition
Let be a finite group. We write
(1)
for the orbit category of .
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:
Its opposite category we call the category of finite-dimensional dual vector G-spaces:
(2)
(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)
The category (3) is denoted in (Triantafillou 82).
Properties
Injective objects
Example
(restriction of vector -spaces to Weyl group representations)
Let any subgroup. Notice that its Weyl group is the automorphism group of its coset space in the orbit category:
(4)
This gives an full subcategory-inclusion
of the delooping category of they Weyl group into the orbit category of (1), and hence a restriction functor
(5)
or more generally
(6)
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 -spaces)
For a subgroup and
a rational finite dimensional left representation of the Weyl group of in , write
for the functor from the -orbit category to rational vector spaces which assigns to a coset space the vector space of homomorphisms of right actions by the Weyl group (4) from the hom-set to the dual vector space equipped with its dual action.
More generally, for
set
This construction extends to a functor right adjoint to the restriction (5):
(Triantafillou 82, (4.1), Golasinski 97a, Lemma 1.1, Scull 08, Def. 2.2, Lemma 2.3)
Proposition
(i) The objects of the form (Def. ) are injective objects in dual vector G-spaces (Def. ).
(ii) Every injective dual vector -space is a direct sum of objects of this form, specifically (see Def. below):
(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 -space)
Let a simplicial set equipped with -action, say that the equivariant PL de Rham complex is the functor on the orbit category
which to a coset space assigns the PL de Rham complex of the -fixed locus .
Then the underlying dual vector G-space
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 (3), its injective envelope is
where
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the direct sum is over conjugacy classes of subgroups, with on the right any one representative of its conjugacy class,
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for the argument of is taken to be all of ,
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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
Example
(orbit category of Z/2Z)
For equivariance group the cyclic group of order 2:
the orbit category looks like this:
(7)
i.e.:
Write
for the two irreducible representations (the trivial representation and the sign representation, respectively) of the Weyl group .
Their induced injective dual vector -spaces, according to Def. , are:
and
Similarly, write
for the unique irrep of the Weyl group .
Its induced injective dual vector -spaces, according to Def. , is:
References
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Georgia Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) (jstor:1999119)
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Marek Golasiński, Componentwise injective models of functors to DGAs, Colloquium Mathematicum, Vol. 73, No. 1 (1997) (dml:21048, pdf)
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Marek Golasiński, Injective models of -disconnected simplicial sets, Annales de l’Institut Fourier, Volume 47 (1997) no. 5, p. 1491-1522 (numdam:AIF_1997__47_5_1491_0)
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Laura Scull, Rational -equivariant homotopy theory, Transactions of the AMS, Volume 354, Number 1, Pages 1-45 2001 (pdf, doi:10.1090/S0002-9947-01-02790-8)
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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)