Algebraic topology computations and …math.univ-lille1.fr/~touze/NotesRecherche/Arolla.pdfAlgebraic...

Algebraic topology computationsand

Representation theory of GLn

Antoine Touze

Universite Paris 13

Arolla 2012

In this talk...

k is a PID (ex : algebraically closed field, Fp or Z).

We will deal with two kinds of functors :

I Ordinary functors, that is :

F : (Free) k-mod→ k-mod .

Huge number of examples.

• ⊗n : V 7→ V ⊗k · · · ⊗k V︸︷︷︸n times

• Sn : V 7→ Sn(V ) = (V⊗n)Sn .

I Strict polynomial functors, that is :

F : Free k-mod→ k-mod .

+ additional algebraic structure.

They come from representation theory of GLn.

• ⊗n and Sn have a canonical structure of strict polynomialfunctors.

In this talk...

• ⊗n : V 7→ V ⊗k · · · ⊗k V︸︷︷︸n times

• Sn : V 7→ Sn(V ) = (V⊗n)Sn .

In this talk...

• ⊗n : V 7→ V ⊗k · · · ⊗k V︸︷︷︸n times

• Sn : V 7→ Sn(V ) = (V⊗n)Sn .

In this talk...

• ⊗n : V 7→ V ⊗k · · · ⊗k V︸︷︷︸n times

• Sn : V 7→ Sn(V ) = (V⊗n)Sn .

In this talk...

• ⊗n : V 7→ V ⊗k · · · ⊗k V︸︷︷︸n times

• Sn : V 7→ Sn(V ) = (V⊗n)Sn .

In this talk...

• ⊗n : V 7→ V ⊗k · · · ⊗k V︸︷︷︸n times

• Sn : V 7→ Sn(V ) = (V⊗n)Sn .

In this talk...

• ⊗n : V 7→ V ⊗k · · · ⊗k V︸︷︷︸n times

• Sn : V 7→ Sn(V ) = (V⊗n)Sn .

In this talk...

• ⊗n : V 7→ V ⊗k · · · ⊗k V︸︷︷︸n times

• Sn : V 7→ Sn(V ) = (V⊗n)Sn .

In this talk...

• ⊗n : V 7→ V ⊗k · · · ⊗k V︸︷︷︸n times

• Sn : V 7→ Sn(V ) = (V⊗n)Sn .

In this talk...

Algebraictopology

Representationsof GLn

Ordinary functors Strict polynomial functors

derived functor of FLF

Ringel dual of FΘF

[Dold-Puppe, Quillen] [Ringel, Donkin, Cha lupnik]

Plan :

I. Functors in algebraic topology.

II. Functors in representation theory of GLn

III. Thm : LF and ΘF coincide.

IV. Some applications.

V. The homology of EML spaces.

In this talk...

Algebraictopology

Ringel dual of FΘF

Plan :

In this talk...

Algebraictopology

Ringel dual of FΘF

Plan :

In this talk...

Algebraictopology

Ringel dual of FΘF

[Dold-Puppe, Quillen]

[Ringel, Donkin, Cha lupnik]

Plan :

In this talk...

Algebraictopology

Ringel dual of FΘF

[Dold-Puppe, Quillen]

[Ringel, Donkin, Cha lupnik]

Plan :

In this talk...

Algebraictopology

Ringel dual of FΘF

Plan :

In this talk...

Algebraictopology

Ringel dual of FΘF

Plan :

I. Functors in algebraic topology (1)

Let A be an abelian group.

I Spaces X (A) constructed from A :

1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :

Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .

I Functors h : Top→ Ab : πi , πsi ,Hi , . . .

I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))

πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !

1. How are they related ?

2. Which ones are isomorphic ?

3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)

I Spaces X (A) constructed from A :1. EML spaces K (A, n),

2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :

Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .

I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)

3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :

Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .

I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .

4. Use functorial constructions :Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .

I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :

Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .

πkM(A, n), HkK (A, n)

, πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !

Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .

πkM(A, n), HkK (A, n), πkΣK (A, n)

, HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !

Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .

πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n)

,πkK (A, n) ∨ K (A, n + 1), . . .What a mess !

Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .

πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .

What a mess !

Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .

πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .

What a mess !

Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .

Derived functors of ordinary functors [Dold-Puppe]

I They help to explain and organize the whole picture

1. Dold Kan correspondence :Normalized Chain functor :N : s(k-mod)→ Ch≥0(k-mod)

is an equivalence of categories, with explicit inverseK : Ch≥0(k-mod)→ s(k-mod) .

N ,K preserve homotopy equivalences.

2. Derived functors :F :

k-mod→ k-mod (Ex : Sd), M ∈ k-mod.

Def : LiF (M; n) =

HiNFKPM [n]Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-modk-modRk : Result does not depend on the choice of PM

Def : LiF (M; n) =

PM [n]Free (or projective) resolution of M, shifted

Free simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-modk-modRk : Result does not depend on the choice of PM

Def : LiF (M; n) =

KPM [n]

Free (or projective) resolution of M, shifted

Free simplicial k-mod, with homology M[n]

Simplicial k-modChain complex of k-modk-modRk : Result does not depend on the choice of PM

Def : LiF (M; n) =

FKPM [n]

Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]

Simplicial k-mod

Chain complex of k-modk-modRk : Result does not depend on the choice of PM

Def : LiF (M; n) =

NFKPM [n]

Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-mod

Chain complex of k-mod

k-modRk : Result does not depend on the choice of PM

Def : LiF (M; n) = HiNFKPM [n]

Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-mod

Rk : Result does not depend on the choice of PM

Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-modk-mod

2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd), M ∈ k-mod.

Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-modk-mod

2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).

LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n])

= πiLF (K (M, n)) [Q]Comments :

I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.

I If F additive, LiF (M, n) = Li−nF (M).

I Definition : [Dold Puppe 1961]

, later generalized [Quillen]

LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)

3. Examples (k = Z, A abelian group) :

1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).

2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).

3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .

4. Curtis spectral sequence (1965) :

E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .

free Lie functor

= πiLF (K (M, n)) [Q]

Comments :

free Lie functor

= πiLF (K (M, n)) [Q]

Comments :

free Lie functor

= πiLF (K (M, n)) [Q]

Comments :

free Lie functor

= πiLF (K (M, n)) [Q]

Comments :

I Definition : [Dold Puppe 1961] , later generalized [Quillen]

free Lie functor

LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n)) [Q]

Comments :

I Definition : [Dold Puppe 1961] , later generalized [Quillen]

free Lie functor

Comments :

free Lie functor

Comments :

free Lie functor

Comments :

free Lie functor

Comments :

free Lie functor

Comments :

free Lie functor

To sum up :

I F : Free k-mod→ k-mod (Ex : Sd).

LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n))

I Derived functors (of Sd , GR, L,. . . ) have topologicalinterpretations

I We want to know more about them !

I 60es-70es : lot of work has been done to compute them[Bott, Bousfield, Curtis, Quillen, Schlessinger . . . ]

But there remains work : for ex, still not clear what LiSd(A, n)

To sum up :

II. Functors in representation theory of GLn (1)

1. Functors → representations of GLn(k)Fk := category of ordinary functors :

F : Free k-mod→ k-mod

Evaluation functor :ev : Fk → GLn(k)-mod

F 7→ F (kn) +ρ :

GLn(k) → GL(F (kn))g 7→ F (g)

2. Representations of GLn,kNow we think of GLn,k as an algebraic group scheme.A representation of GLn,k is

M ∈ k-mod + ρ : GLn,k → GLk(M)morphism of algebraic group schemes

Forgetful functor : Alg. Group Schemes→ Groups

(G ,OG ) 7→ G

Forget that ρ is a morphism of algebraic group schemes :

U : GLn,k-mod → GLn(k)-mod(M, ρ) 7→ (M, ρ)

F : Free k-mod→ k-modEvaluation functor :ev : Fk → GLn(k)-mod

F 7→ F (kn) +ρ :

GLn(k) → GL(F (kn))g 7→ F (g)

(G ,OG ) 7→ G

F 7→ F (kn) +ρ : GLn(k) → GL(F (kn))g 7→ F (g)

(G ,OG ) 7→ G

2. Representations of GLn,kNow we think of GLn,k as an algebraic group scheme.

A representation of GLn,k isM ∈ k-mod + ρ : GLn,k → GLk(M)

morphism of algebraic group schemes

(G ,OG ) 7→ G

Forgetful functor : Alg. Group Schemes→ Groups(G ,OG ) 7→ G

(G ,OG ) 7→ G

Examples of representations of GLn,k :

k = Fp.

(a) Id : GLn,k → GLn,k is algebraic group scheme morphism.

(kn, Id) is a representation of GLn,k

(b) The polynomial formula ρFrob GLn,k → GLn,k[ai ,j ] 7→ [ai ,j

p]defines an algebraic group scheme morphism.

(kn, ρFrob) is a representation of GLn,k

(c) Observe that ρFrob and Id :I are different as morphisms of algebraic groups,I coincide as maps between the sets GLn(k)→ GLn(k).

I (kn, ρFrob) 6= (kn, Id) in GLn,k-modI U(kn, ρFrob) = U(kn, Id) in GLn(k)-mod

k = Fp.

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3. Strict polynomial functors

strict polynomial functors =

GLn,k-mod

Fkev // GLn(k)-mod

strict polynomial functors = Pkev //

GLn,k-mod

Fkev // GLn(k)-mod

Def : Strict polynomial functor = pair (F , FM,N) with

I F ∈ FkI For all free k-mod M,N, polynomial

FM,N : Homk(M,N)→ Homk(F (M),F (N))

such that FM,N(f ) = F (f )

(+ technical axiom)

Def : A strict polynomial functor is homogeneous of degree d if allthe FM,N are homogeneous of degree d .

Ex : Sd , ⊗d , Ld are canonically strict polynomial functors,homogeneous of degree d .

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strict polynomial functors = GLn,k-mod

Fkev // GLn(k)-mod

GLn,k-mod

Fkev // GLn(k)-mod

(+ technical axiom)

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Fkev // GLn(k)-mod

GLn,k-mod

Fkev // GLn(k)-mod

(+ technical axiom)

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Fkev // GLn(k)-mod

GLn,k-mod

Fkev // GLn(k)-mod

I F ∈ Fk

I For all free k-mod M,N, polynomial

(+ technical axiom)

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Fkev // GLn(k)-mod

GLn,k-mod

Fkev // GLn(k)-mod

(+ technical axiom)

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Fkev // GLn(k)-mod

GLn,k-mod

Fkev // GLn(k)-mod

such that FM,N(f ) = F (f ) (+ technical axiom)

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Fkev // GLn(k)-mod

GLn,k-mod

Fkev // GLn(k)-mod

(+ technical axiom)

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Fkev // GLn(k)-mod

GLn,k-mod

Fkev // GLn(k)-mod

(+ technical axiom)

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GLn,k-mod

Fkev // GLn(k)-mod

Def : Strict polynomial functor = pair (F , FM,N) (with F ∈ Fk)Ex : Sd , ⊗d , Ld (homogeneous of degree d).

Notation : Pd ,k = full subcategory of Pkwith objects homogeneous functors of degree d .

I Strict polynomial functors are interesting because :

Thm : [Friedlander-Suslin] If n ≥ d , iso :

Ext∗Pd,k(F ,G ) ' Ext∗GLn,k(F (kn),G (kn))

Allows explicit ExtGLn,k-computations !

Ingredient for finitegeneration theorems [FS, 97] [T, Van der Kallen, 2010].

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GLn,k-mod

Fkev // GLn(k)-mod

Def : Strict polynomial functor = pair (F , FM,N) (with F ∈ Fk)Ex : Sd , ⊗d , Ld (homogeneous of degree d).Notation : Pd ,k = full subcategory of Pk

with objects homogeneous functors of degree d .

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GLn,k-mod

Fkev // GLn(k)-mod

Allows explicit ExtGLn,k-computations ! Ingredient for finitegeneration theorems [FS, 97] [T, Van der Kallen, 2010].

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GLn,k-mod

Fkev // GLn(k)-mod

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GLn,k-mod

Fkev // GLn(k)-mod

Allows explicit ExtGLn,k-computations ! Ingredient for finitegeneration theorems [FS, 97] [T, Van der Kallen, 2010].

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Algebraictopology

Representationsof group scheme GLn,k

Fk = ordinary functors :F : Free k-mod→ k-mod

Sd , ⊗d , Ld ,. . .

Pd ,k = strict polynomial functors :F ∈ Fk + additional structure

Sd , ⊗d , Ld ,. . .

derived functors [DP,Q] :LiF (M, n)

related to alg. top.computations

4. Ringel Duality Θ

I [Ringel, 91] repres. finite dim. algebrasI [Donkin, 93] the GLn,k-mod caseI [Cha lupnik, 2008] translated to Pd ,k

Ringel duality linked with theory of tilting modules

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Algebraictopology

Sd , ⊗d , Ld ,. . .

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Algebraictopology

Sd , ⊗d , Ld ,. . .

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Algebraictopology

Sd , ⊗d , Ld ,. . .

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Algebraictopology

Sd , ⊗d , Ld ,. . .

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Algebraictopology

Sd , ⊗d , Ld ,. . .

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A) Functional Homs

Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k

Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k

If G fixed, functional Hom yields :

Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )

B) Case G = Λd (exterior power).

Def : Ringel duality operatorΘ =R

Hom(Λd ,−) :

Pd ,k →

(left exact)

Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.

Thm : [RDC] Θ is an equivalence of categories.Ex : Θ(Sd) = Λd ,

Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)Θ(Γd) = big complex !

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A) Functional Homs

Hom(Λd ,−) :

Pd ,k →

(left exact)

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A) Functional Homs

Hom(Λd ,−) :

Pd ,k →

(left exact)

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A) Functional Homs

Hom(Λd ,−) :

Pd ,k →

Pd ,k (left exact)

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A) Functional Homs

Hom(Λd ,−) :

Pd ,k →

Pd ,k (left exact)

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A) Functional Homs

Def : Ringel duality operatorΘ =

RHom(Λd ,−) : DbPd ,k → DbPd ,k

(left exact)

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A) Functional Homs

Def : Ringel duality operatorΘ =RHom(Λd ,−) : DbPd ,k → DbPd ,k

(left exact)

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A) Functional Homs

(left exact)

Thm : [RDC] Θ is an equivalence of categories.

Ex : Θ(Sd) = Λd ,

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A) Functional Homs

(left exact)

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A) Functional Homs

(left exact)

Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)

Θ(Γd) = big complex !

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A) Functional Homs

(left exact)

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III. ΘF and LF coincide (1)

Algebraictopology

Sd , ⊗d , Ld ,. . .

Pd ,k = strict polynomial functorsF ∈ Fk + additional structure

Sd , ⊗d , Ld ,. . .

Ringel duals :Θ(F ) ∈ DbPd ,k

(alg. top. computations) (tilting modules)

Prop : If M is free k-mod, F ∈ Pd ,k then M 7→ LiF (M, n) ∈ Pd ,k.

Thm : [T] If M is free k-mod, F ∈ Pd ,k :

Lnd−iF (M; n) ' H i (ΘnF )(M)

Rq : Stronger statement on the level of derived categories.Cor : Ld−iF (M, 1) = ExtiPd,k

(Λd ,FM),

L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)

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Algebraictopology

Sd , ⊗d , Ld ,. . .

(Λd ,FM),

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Algebraictopology

Sd , ⊗d , Ld ,. . .

(Λd ,FM),

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Algebraictopology

Sd , ⊗d , Ld ,. . .

Rq : Stronger statement on the level of derived categories.

Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),

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Algebraictopology

Sd , ⊗d , Ld ,. . .

(Λd ,FM),

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Algebraictopology

Sd , ⊗d , Ld ,. . .

(Λd ,FM),

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Why is this theorem interesting ?

Left hand side Right hand side

Simplicial methodsIntuition from alg. top.

Representation theory methods :• Highest weight categories• Tilting modules• Block theory. . .

Different insights on computations.Well-known computations on the right hand side are unknown onthe left hand side (and vice versa).

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Representation theory methods :• Highest weight categories

• Tilting modules• Block theory. . .

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Representation theory methods :• Highest weight categories• Tilting modules

• Block theory. . .

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The proof

Easy. Ingredients :

1. Classical computation : L∗Sd(M, 1) ' Λd(M)[d ]

equivalent forms :I Koszul dual of S(M) is Λ(M),

I Ext∗S(M)(k,k) = Λ(M[1]),I ΩCP∞ ' S1,

2. Standard simplicial algebra (EZ-thm. . .)3. Elementary homological algebra in Pd ,k (Yoneda lemma).

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The proof

Ingredients :

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The proof

Easy. Ingredients :

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The proof

Easy. Ingredients :

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The proof

Easy. Ingredients :

equivalent forms :I Koszul dual of S(M) is Λ(M),I Ext∗S(M)(k,k) = Λ(M[1]),

I ΩCP∞ ' S1,

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The proof

Easy. Ingredients :

equivalent forms :I Koszul dual of S(M) is Λ(M),I Ext∗S(M)(k,k) = Λ(M[1]),I ΩCP∞ ' S1,

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The proof

Easy. Ingredients :

2. Standard simplicial algebra (EZ-thm. . .)

3. Elementary homological algebra in Pd ,k (Yoneda lemma).

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The proof

Easy. Ingredients :

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IV. Applications (1)

1. Application to GLn,k-modules.

A) Plethysm problem.

Let M ∈ GLn,k-mod.Plethysm = GLn,k-mod of the form F (M), F functor.

Difficult Problem : understand F (M).• Composition factors ?• Extensions Ext∗GLn,k

(N,F (M)) ?

B) Translation to Pk :Plethysm = functor of the form F GProblem : understand F G

Example : Sn ⊗d .Question : Can we compute Ext∗Pnd,k

(H,Sn ⊗d) ?

Rk : no injective resolution of Sn ⊗d is known,already HomPnd,k(H,Sn ⊗d) is not easy.

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(N,F (M)) ?

(H,Sn ⊗d) ?

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(N,F (M)) ?

(H,Sn ⊗d) ?

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(N,F (M)) ?

B) Translation to Pk :

Plethysm = functor of the form F GProblem : understand F G

(H,Sn ⊗d) ?

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(N,F (M)) ?

(H,Sn ⊗d) ?

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(N,F (M)) ?

Example : Sn ⊗d .

Question : Can we compute Ext∗Pnd,k(H,Sn ⊗d) ?

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(N,F (M)) ?

(H,Sn ⊗d) ?

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(N,F (M)) ?

(H,Sn ⊗d) ?

Rk : no injective resolution of Sn ⊗d is known,already HomPnd,k(H, Sn ⊗d) is not easy.

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Question : Can we compute Ext∗Pnd,k(H

,Sn ⊗d) ?

C) Translation in derived functors

Question : Can we compute L∗(Sn ⊗d)(k, 1) ?

I Recall L∗(Sn ⊗d)(k, 1) = homology of

N (Sn ⊗d)K(k[1])NSn(K(k[1])⊗d)

I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S

n ⊗d)(k, 1) = L∗Sn(k, d)

D) In general :

Thm : F ∈ Pn,k, G ∈ Pd ,k,

HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)

ExtiPnd,k(Λnd ,F GM) ' (HiΘ

dF ) (H0ΘG ) (M)

Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0

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Question : Can we compute Ext∗Pnd,k(

Λnd ,Sn ⊗d) ?

n ⊗d)(k, 1) = L∗Sn(k, d)

D) In general :

dF ) (H0ΘG ) (M)

- 18/1

Λnd ,Sn ⊗d) ?

n ⊗d)(k, 1) = L∗Sn(k, d)

D) In general :

dF ) (H0ΘG ) (M)

- 18/1

Λnd ,Sn ⊗d) ?

N (Sn ⊗d)K(k[1])

NSn(K(k[1])⊗d)

n ⊗d)(k, 1) = L∗Sn(k, d)

D) In general :

dF ) (H0ΘG ) (M)

- 18/1

Λnd ,Sn ⊗d) ?

N (Sn ⊗d)K(k[1])

NSn(K(k[1])⊗d)

n ⊗d)(k, 1) = L∗Sn(k, d)

D) In general :

dF ) (H0ΘG ) (M)

- 18/1

Λnd ,Sn ⊗d) ?

N (Sn ⊗d)K(k[1])

NSn(K(k[1])⊗d)

I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).

I We get : L∗(Sn ⊗d)(k, 1) = L∗S

n(k, d)

D) In general :

dF ) (H0ΘG ) (M)

- 18/1

Λnd ,Sn ⊗d) ?

N (Sn ⊗d)K(k[1])

NSn(K(k[1])⊗d)

n ⊗d)(k, 1) = L∗Sn(k, d)

D) In general :

dF ) (H0ΘG ) (M)

- 18/1

Λnd ,Sn ⊗d) ?

N (Sn ⊗d)K(k[1])

NSn(K(k[1])⊗d)

n ⊗d)(k, 1) = L∗Sn(k, d)

D) In general :

dF ) (H0ΘG ) (M)

- 18/1

Λnd ,Sn ⊗d) ?

N (Sn ⊗d)K(k[1])

NSn(K(k[1])⊗d)

n ⊗d)(k, 1) = L∗Sn(k, d)

D) In general :

Thm : F ∈ Pn,k, G ∈ Pd ,k, HiΘ(G ) = 0 if i 6= 0

(Sd , Λd , ⊗d , Schur functors. . .)ExtiPnd,k

(Λnd ,F GM) ' (HiΘdF ) (H0ΘG ) (M)

- 18/1

Λnd ,Sn ⊗d) ?

N (Sn ⊗d)K(k[1])

NSn(K(k[1])⊗d)

n ⊗d)(k, 1) = L∗Sn(k, d)

D) In general :

Thm : F ∈ Pn,k, G ∈ Pd ,k, HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)

dF ) (H0ΘG ) (M)

- 18/1

Λnd ,Sn ⊗d) ?

N (Sn ⊗d)K(k[1])

NSn(K(k[1])⊗d)

n ⊗d)(k, 1) = L∗Sn(k, d)

D) In general :

dF ) (H0ΘG ) (M)

- 18/1

Λnd ,Sn ⊗d) ?

N (Sn ⊗d)K(k[1])

NSn(K(k[1])⊗d)

n ⊗d)(k, 1) = L∗Sn(k, d)

D) In general :

dF ) (H0ΘG ) (M)

- 19/1

2. Application to derived functors.k = Z.Let Y d = kernel of Λd−1 ⊗ Λ1 mult−−−→ Λd Y d ∈ Pd ,Z.

The Y ds appear in the study of the Curtis spectral sequence.

Thm : If p 6 |d , then the p-torsion part of L∗Yd(Z, 1) is zero.

Proof :I We have to prove L∗Y

d(Z, 1)⊗ Fp = 0.I By univ. coeff. thm, it suffices to prove L∗Y

(Fp, 1) = 0.

I So it suffices to prove Ext∗Pd,Fp(Λd ,Y d

Fp) = 0.

I This follows without any computation from block theory.

• A abelian category (+finiteness hyp),blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]• Λd → (1d), Y d → (2, 1d−2).

- 19/1

(Fp, 1) = 0.

Fp) = 0.

- 19/1

d(Z, 1)⊗ Fp = 0.

I By univ. coeff. thm, it suffices to prove L∗YdFp

(Fp, 1) = 0.

Fp) = 0.

- 19/1

(Fp, 1) = 0.

Fp) = 0.

- 19/1

(Fp, 1) = 0.

Fp) = 0.

- 19/1

(Fp, 1) = 0.

Fp) = 0.

- 19/1

(Fp, 1) = 0.

Fp) = 0.

I This follows without any computation from block theory.• A abelian category (+finiteness hyp),

blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.

• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]• Λd → (1d), Y d → (2, 1d−2).

- 19/1

(Fp, 1) = 0.

Fp) = 0.

blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]

• Λd → (1d), Y d → (2, 1d−2).

- 19/1

(Fp, 1) = 0.

Fp) = 0.

blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]• Λd → (1d), Y d → (2, 1d−2).

- 20/1

V. The homology of EML spaces (1)

Work in progress with. L. Breen and R. Mikhailov.

k = Z, A is an abelian group.

1. LiSd(A, n) important for representation theory of GLn,Z

(A free abelian), ex :I ExtiPd,Z

(Λd , ΓdA) = L3d−iS

d(A, 3)

I ExtiPnd,Z(Λnd ,Sn ⊗d) = Lnd−iS

n(A⊗d , d)

I is one of the less complicated functor,I appears everywhere,

3. Thm : [Dold-Puppe] H∗K (A, n) ' L∗S(A, n)

(A free abelian group, otherwise iso up to filtration)

We want to compute L∗Sd(A, n) !

Easy cases : L∗Sd(A, 1), L∗S

d(A, 2),

in general ? ?

- 20/1

d(A, 3)

n(A⊗d , d)

d(A, 2),

in general ? ?

- 20/1

d(A, 3)

n(A⊗d , d)

d(A, 2),

in general ? ?

- 20/1

d(A, 3)

n(A⊗d , d)

d(A, 2),

in general ? ?

- 20/1

d(A, 3)

n(A⊗d , d)

3. Thm : [Dold-Puppe] H∗K (A, n) ' L∗S(A, n)(A free abelian group, otherwise iso up to filtration)

d(A, 2),

in general ? ?

- 20/1

d(A, 3)

n(A⊗d , d)

d(A, 2),

in general ? ?

- 20/1

d(A, 3)

n(A⊗d , d)

d(A, 2),

in general ? ?

- 20/1

d(A, 3)

n(A⊗d , d)

d(A, 2), in general ? ?

- 21/1

Problem : Compute L∗Sd(A, n).

L∗S(A, n) ' H∗K (A, n), and we know homology of EML spaces.

”everything is computed in the Cartan Seminar [54]”

”. . .but not under an easy-to-use form !”

[Cartan seminar, tome 7, 1954, expose. 11] :

Enormous DGA H∗Xp(A, n)φp−→ H∗K (A, n)Enormous algebra

I φp surjects on p-primary part.I kerφp is generated by 4 pages of relations.

With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations, but :

I Hi (A, n) is finitely generated abelian group !

I Hi (A, n) '⊕

d≥0 LiSd(A, n), how do we get the summands ?

- 21/1

Enormous DGA H∗Xp(A, n)φp−→ H∗K (A, n)Enormous algebra

I Hi (A, n) '⊕

- 21/1

Enormous DGA

Xp(A, n)

φp−→ H∗K (A, n)Enormous algebra

I Hi (A, n) '⊕

- 21/1

Enormous DGA

H∗Xp(A, n)

φp−→ H∗K (A, n)

Enormous algebra

I Hi (A, n) '⊕

- 21/1

Enormous DGA

H∗Xp(A, n)φp−→ H∗K (A, n)Enormous algebra

I φp surjects on p-primary part.

I kerφp is generated by 4 pages of relations.

I Hi (A, n) '⊕

- 21/1

Enormous DGA

I Hi (A, n) '⊕

- 21/1

Enormous DGA

With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations

, but :

I Hi (A, n) '⊕

- 21/1

Enormous DGA

I Hi (A, n) '⊕

- 21/1

Enormous DGA

I Hi (A, n) is finitely generated abelian group !I Hi (A, n) '

⊕d≥0 LiS

d(A, n), how do we get the summands ?

- 21/1

”everything is computed in the Cartan Seminar [54]””. . .but not under an easy-to-use form !”

Enormous DGA

I Hi (A, n) is finitely generated abelian group !I Hi (A, n) '

⊕d≥0 LiS

d(A, n), how do we get the summands ?

- 22/1

In 1954, Cartan could not use :

1. theory of derived functors,2. the fact that H∗K (A, n) ' LiS

d(A, n) is strict polynomial,3. representation theory of GLn,Z.

Hope :Using these additional tools, we hope to get usable results.(ex : complete results for LiS

d(A, 3))

Thank You !

- 22/1

1. theory of derived functors,

2. the fact that H∗K (A, n) ' LiSd(A, n) is strict polynomial,

3. representation theory of GLn,Z.

d(A, 3))

Thank You !

- 22/1

d(A, n) is strict polynomial,

3. representation theory of GLn,Z.

d(A, 3))

Thank You !

- 22/1

d(A, 3))

Thank You !

- 22/1

d(A, 3))

Thank You !

- 22/1

d(A, 3))

Thank You !

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