homology representations of the...

Homology Representations of the SymmetricGroups1

Netanel Friedenberg

October 8, 2014

1This material is based upon work supported by the National ScienceFoundation under agreement No. DMS-1055897. Any opinions, findings andconclusions or recommendations expressed in this material are those of theauthors and do not necessarily reflect the views of the National ScienceFoundation

Group Actions

An action of a group G on a set A is a group homomorphismϕ : G → SA, or

a multiplication G × A→ A, (g , x) 7→ gx such that ∀x ∈ Aand ∀g , h ∈ G

1Gx = x and(gh)x = g(hx).

Group Actions

Group Representations

Representation of G : A group homomorphismX : G → GLd(C).

Equivalently, G -module: A vector space V , with amultiplication G × V → V , (g , v) 7→ v, such that(∀v,w ∈ V )(∀c , d ∈ C)(∀g , h ∈ G ),

1Gv = v.(gh)v = g(hv).g(cv + dw) = c(gv) + d(gw).

An Example: Permutation Representations

Say G acts on A. The permutation representation associated withthis action is:

Vector space V = CA.

Multiplication: g(r1a1 + · · ·+ rmam) = r1(ga1) + · · ·+ rm(gam)(ri ∈ C, ai ∈ A, g ∈ G ).

Homology

Chain complex: (C·, ∂·) a sequence

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

of vector spaces Ck (“chain spaces”) and lineartransformations ∂k : Ck → Ck−1 (“boundary maps”) suchthat ∂k ◦ ∂k+1 = 0 (∀k ∈ Z).

Often omit subscripts: ∂2 = 0.

∂k ◦ ∂k+1 = 0 ⇐⇒ Im(∂k+1) ⊂ Ker(∂k).

Homology spaces: Hk :=Ker(∂k)

Im(∂k+1)

Homology

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

∂k ◦ ∂k+1 = 0 ⇐⇒ Im(∂k+1) ⊂ Ker(∂k).

Im(∂k+1)

Homology

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

∂k ◦ ∂k+1 = 0 ⇐⇒ Im(∂k+1) ⊂ Ker(∂k).

Im(∂k+1)

Homology

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

∂k ◦ ∂k+1 = 0 ⇐⇒ Im(∂k+1) ⊂ Ker(∂k).

Im(∂k+1)

Example: Homology of a triangle

We assign a chain complex to the boundary of triangle ∆ABC :

Let C1 = span(AB,BC ,CA), C0 = span(A,B,C ), andC−1 = span(∅).

Boundary map: “oriented boundary”. That is,

∂1(AB) = B − A, ∂1(BC ) = C − B, ∂1(CA) = A− C ,∂0(A) = ∂0(B) = ∂0(C ) = 1 · ∅,∂−1(∅) = 0.

∂1(AB) = B − A, ∂1(BC ) = C − B, ∂1(CA) = A− C ,

∂0(A) = ∂0(B) = ∂0(C ) = 1 · ∅,∂−1(∅) = 0.

∂1(AB) = B − A, ∂1(BC ) = C − B, ∂1(CA) = A− C ,∂0(A) = ∂0(B) = ∂0(C ) = 1 · ∅,

∂−1(∅) = 0.

It is easy to check that in this case we have

ker(∂1) = span(AB + BC + CA)Im(∂1) = {aA + bB + cC ∈ C0|a + b + c = 0} = ker(∂0)Im(∂0) = C−1 = ker(∂−1)

So the homology spaces are H1∼= C, H0

∼= 0, and H−1 ∼= 0.

Generalization: Simplicial homology of a simplicial complex.

Further generalization (via category theory): Simplicial sets.

∼= 0, and H−1 ∼= 0.

Homology Representations

Say G is a group and

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

is a chain complex where the chain spaces are also G -modules.

Suppose further: multiplication by group elements commuteswith boundary maps.

That is, ∂(gv) = g∂(v).

Then the homology spaces become G -modules in a naturalway.

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

Chain complex from before, based on the triangle.S{A,B,C} ∼= S3 acts on the triangle by permuting its vertices.

So S3 acts on the chain spaces.

H1 = span(AB + BC + CA). While it seems that S3 actstrivially, we are concerned with orientation, so, for example,BA = −AB.

It turns out thatg(AB + BC + CA) = sign(g)(AB + BC + CA), where

sign(g) =

{1 if g is an even permutation

−1 if g is an odd permutation.

sign(g) =

N-Complexes

In a paper in 1996, Kapranov introduced the notion of anN-complex.

An N-complex (C·, ∂·) is a sequence

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

of vector spaces Ck and linear transformations∂k : Ck → Ck−1 such that ∂N = 0.

So a 2-complex is a chain complex.

Choose k ∈ Z, 1 ≤ i < N. Then

· · · ∂ i

−→ Ck+N−i∂N−i

−→ Ck∂ i

−→ Ck−i∂N−i

−→ · · ·

is a chain complex. The homology at Ck is

Hk,i =ker(∂ i : Ck → Ck−i )

Im(∂N−i : Ck+N−i → Ck)

N-Complexes

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

· · · ∂ i

−→ Ck∂ i

−→ · · ·

N-Complexes

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

· · · ∂ i

−→ Ck∂ i

−→ · · ·

N-Complexes

· · · ∂n+1−→ Cn∂n−→ Cn−1

∂n−1−→ · · · ∂1−→ C0∂0−→ C−1

∂−1−→ · · ·

· · · ∂ i

−→ Ck∂ i

−→ · · ·

N-Complexes

Kapranov shows that given a simplicial set, and ω an Nth rootof unity, using powers of ω as weights in the sums defining theboundary maps yields an N-complex associated to thesimplicial set.

In fact, the usual chain complex associated to the simplicialset is just the case N = 2, ω = −1.

N-Complexes

Kapranov shows that given a simplicial set, and ω an Nth rootof unity, using powers of ω as weights in the sums defining theboundary maps yields an N-complex associated to thesimplicial set.

In fact, the usual chain complex associated to the simplicialset is just the case N = 2, ω = −1.

Word Complexes

Inspired by Kapranov’s work, we constructed the followingN-complexes. Fix ω, an Nth root of unity, and n ∈ N

Wk , spanned by k-tuples from {1, . . . , n}.Boundary map:

∂k((a1, . . . , ak)) =k∑

ωi−1(a1, . . . , ai , . . . , ak) (ai indicates

that ai has been removed from the list).

(W·, ∂·) is an N-complex, the (total) word complex.

Ik , subspace of Wk spanned by k-tuples with no repeats.Then (I·, ∂·) is also an N-complex, the injective word complex.

Word Complexes

Wk , spanned by k-tuples from {1, . . . , n}.

Boundary map:

∂k((a1, . . . , ak)) =k∑

Word Complexes

∂k((a1, . . . , ak)) =k∑

Word Complexes

∂k((a1, . . . , ak)) =k∑

Word Complexes

∂k((a1, . . . , ak)) =k∑

Word Complexes

Sn acts on the collection of (injective) k-tuples of thenumbers 1, . . . , n: Wk and Ik are permutation modules for Sn.

This action of Sn commutes with the boundary maps ∂k , soHk,i (W ) and Hk,i (I ) are Sn-modules as well.

Word Complexes

Sn acts on the collection of (injective) k-tuples of thenumbers 1, . . . , n: Wk and Ik are permutation modules for Sn.

This action of Sn commutes with the boundary maps ∂k , soHk,i (W ) and Hk,i (I ) are Sn-modules as well.

Total Word Complex

Theorem:

The complex (W·, ∂·) has trivial homology. That is, forall k and all 0 < i < N, Hk,i (W ) = 0.

Total Word Complex

Theorem: The complex (W·, ∂·) has trivial homology. That is, forall k and all 0 < i < N, Hk,i (W ) = 0.

Injective Word Complex

Conjecture: For the injective word complex, all of the homology is“concentrated towards the top”.

Specifically, we conjecture thatfor any of the chain complexes

· · · ∂ i

−→ Ik+N−i∂N−i

−→ Ik∂ i

−→ Ik−i∂N−i

−→ · · · ,

all of the homology spaces except the top one are the zero space,unless the sequence starts

In∂−→ In−1

∂N−1

−→ · · · ,

in which case the top two homology spaces may be nontrivial.

Conjecture: For the injective word complex, all of the homology is“concentrated towards the top”. Specifically, we conjecture thatfor any of the chain complexes

· · · ∂ i

−→ Ik+N−i∂N−i

−→ Ik∂ i

−→ Ik−i∂N−i

−→ · · · ,

all of the homology spaces except the top one are the zero space,unless the sequence starts

In∂−→ In−1

∂N−1

−→ · · · ,

in which case the top two homology spaces may be nontrivial.

Theorem: If the homology is concentrated as conjectured, then thecollection of these homology modules for all n breaks up naturallyinto sequences of representations (Xn)∞n=1 (where each Xn is arepresentation of Sn) that are what Stanley, in his EnumerativeCombinatorics, calls elementary sequences. Moreover, the caseN = 2, ω = −1 gives exactly the sequence of representations thathe gives as an example there.

Current Work

We are currently trying to use methods of algebraic discreteMorse theory as well as methods involving spectral sequencesto prove our conjecture about the concentration of thehomology of the injective word complex.

We are also using the zig-zag lemma, applied to the triple I ,W , and the quotient complex W /I , to convert the conjectureto an equivalent conjecture about W /I , the “non-injectiveword complex”, which seems to behave better with respect tothe manipulations necessary for the algebraic discrete Morsetheory, and opens up more possibilities for the use of spectralsequences.

Current Work

Future work

We can also assign N-complexes analogous to W and I to anysimplicial complex. We are particularly interested in whether thereare connections between these homology spaces and the usualsimplicial homology for a given simplicial complex.

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