On the Homology of the Real Complement of the k -Parabolic - - PowerPoint PPT Presentation

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On the Homology of the Real Complement of the k -Parabolic - - PowerPoint PPT Presentation

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On the Homology of the Real Complement of the k -Parabolic Arrangement Christopher Severs 1 Jacob A. White 2 1 Mathematical Sciences Research Institute Reykjavik University 2 Arizona State University Mathematical Sciences Research Institute

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On the Homology of the Real Complement of the k-Parabolic Arrangement

Christopher Severs1 Jacob A. White2

1Mathematical Sciences Research Institute → Reykjavik University 2Arizona State University → Mathematical Sciences Research Institute

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Outline

1

Background

2

The k-Parabolic Arrangement

3

Cellular Model

4

Discrete Morse Theory

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Subspace arrangements A - is a collection of linear subspaces (over Rn). Assume ∩X∈A X = 0. The complement M(A ) = Rn − ∪X∈A X. Primary interest: Study homology of M(A ). Note: homology groups ’measure’ i-dimensional holes in a space. Has surprising application in complexity theory (and linear decision trees) - Bj¨

  • rner, Lov`

asz, Yao

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

The k-equal arrangement Definition The k-equal arrangment, An−1,k consists of subspaces (of Rn): xi1 = xi2 = ... = xik, for all distinct indices 1 ≤ i1 < ... < ik ≤ n When k=2 we recover the Braid hyperplane arrangement.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

The k-equal arrangement Definition The k-equal arrangment, An−1,k consists of subspaces (of Rn): xi1 = xi2 = ... = xik, for all distinct indices 1 ≤ i1 < ... < ik ≤ n When k=2 we recover the Braid hyperplane arrangement. In 1995, Bj¨

  • rner and Welker computed the homology of M(An,k).

Homology groups are torsion free. Hi is trivial unless i = t(k − 2), some 0 ≤ t ≤ n

k .

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

A type B and D k-equal arrangement In 1996, Bj¨

  • rner and Sagan defined type B and D analogues.

Definition The Dn,k arrangement consists of subspaces (of Rn): ±xi1 = ±xi2 = ... = ±xik, 1 ≤ i1 < · · · < ik ≤ n The Bn,k,h arrangement consists of Dn,k plus subspaces: xi1 = . . . = xih = 0, 1 ≤ i1 < . . . < ih ≤ n They computed the homology of Bn,k,h. In 2000, Kozlov-Feichtner computed the homology of Dn,k, k > n

2.

Homology groups are torsion free

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

The usual approach Usual approach to finding homology of M(A ):

1

Apply Goresky MacPherson formula (to reduce problem to studying homology of ∆(L (A ))

2

Show that ∆(L (A )) is shellable.

3

Deduce that Hi(∆(L (A ))) is torsion-free, get Betti numbers.

4

Obtain results for M(A ).

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

The usual approach Usual approach to finding homology of M(A ):

1

Apply Goresky MacPherson formula (to reduce problem to studying homology of ∆(L (A ))

2

Show that ∆(L (A )) is shellable.

3

Deduce that Hi(∆(L (A ))) is torsion-free, get Betti numbers.

4

Obtain results for M(A ). NOTE THAT WE ARE NOT USING THIS APPROACH!

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Primary goals

1

Generalize k-equal arrangement to any finite reflection group W.

2

Construct a cellular model for the complement.

3

Apply discrete Morse theory to our cellular model.

4

Obtain results regarding homology and Betti numbers.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Outline

1

Background

2

The k-Parabolic Arrangement

3

Cellular Model

4

Discrete Morse Theory

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Reflection group Now we begin our generalization. W - an irreducible finite real reflection group acting on Rn. S = set of simple reflections associated to W. R = {wsw−1 : w ∈ W, s ∈ S} is the set of all reflections

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Coxeter Arrangement for W Definition The Coxeter arrangement is given by hyperplanes Hr = {x ∈ Rn : rx = x} for each r ∈ R.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Definitions for Galois correspondence Definition of k-parabolic arrangement involves parabolic subgroups. Definition G ⊆ W is parabolic if ∃I ⊆ S, w ∈ W such that G =< wIw−1 >. G is k-parabolic if G is irreducible and of rank k − 1.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Definitions for Galois correspondence Definition of k-parabolic arrangement involves parabolic subgroups. Definition G ⊆ W is parabolic if ∃I ⊆ S, w ∈ W such that G =< wIw−1 >. G is k-parabolic if G is irreducible and of rank k − 1. For G ⊂ W, let Fix(G) = {x ∈ Rn : wx = x, ∀w ∈ G} For subspace X ⊂ Rn, let Gal(X) = {w ∈ W : wx = x, ∀x ∈ X}

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Galois Correspondence P(W) - poset of all parabolic subgroups of W ordered by inclusion. L (W) - poset of all ∩s of hyperplanes of H (W), ordered by ⊇. Note: Posets are both lattices. Theorem (Barcelo and Ihrig, 1999) P(W) ∼ = L (W) via G → Fix(G) Gal(X) ← X We use this “Galois correspondence”. But first we give an example for A3 = S4.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Example of correspondence W = S4 L (S4) P(S4)

1234 1/2/3/4 e <12> <34> 23<12>23 34<23>34 24<12>24 <23> 12/34 134/2 123/4 13/24 124/3 1/234 14/23 1/23/4 14/2/3 1/24/3 13/2/4 12/3/4 1/2/34 13<12,34>13 <23,34> 34<12,23>34 23<12,34>23 <12,23> 12<23,34>12 <12,34> <12,23,34>

Example: 14/23 ↔< (1, 4), (2, 3) >= (1, 3) < (1, 2), (3, 4) > (1, 3) Example: 134/2 ↔< (1, 3), (3, 4) >= (1, 2) < (2, 3), (3, 4) > (1, 2)

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Definition of the k-parabolic arrangement Definition Pn,k(W) = the collection of all k-parabolic subgroups of W. The k-parabolic arrangement Wn,k is the collection of subspaces Fix(G), G ∈ Pn,k(W)

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Examples of the k-parabolic arrangement We recover some of the previous examples. W An Bn Dn(k = 3) Dn(k > 3) W(k = 2) Wn,k An,k Bn,k,k−1 Dn,3 Bn,k,k−1 H (W) ∴ The k-equal arrangement has been generalized.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Outline

1

Background

2

The k-Parabolic Arrangement

3

Cellular Model

4

Discrete Morse Theory

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

The Coxeter Complex and Permutahedron H (W) ∩ Sn−1 induces the Coxeter complex C (W). Perm(W), the type W Permutahedron, is a polytope with face poset dual of C (W). Faces of Perm(W) correspond to cosets uWI, ordered by inclusion. Note: WI =< I >, for I ⊂ S.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Picture Wn,k H (W) Ck(W) C (W) Permk(W) Perm(W)

k-parabolic subgroups → all parabolic subgroups (under ⊆) {wWI ⊇ k-parabolic cosets} → all standard parabolic cosets (under ⊇) {wWI ⊇ k-parabolic cosets} → all standard parabolic cosets (under ⊆)

∩Sn−1 dualize ∩Sn−1 dualize ∴ M(Wn,k) is equivalent to removing Permk(W) from Perm(W).

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

The subcomplex Permk(W) Permk(W) - subcomplex of W-Permutahedron, minimal nonfaces correspond to cosets wWI, such that WI is k-parabolic. Proposition M(Wn,k) is homotopy equivalent to Permk(W). Proposition uses techniques of Bj¨

  • rner and Ziegler.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Example with B3,3 H (B3) B3-Permutahedron B3,3

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Conclusion - Cellular Model ∴ We have constructed a cellular model for M(Wn,k). Homology of Permk(W) = Homology of M(Wn,k). Moreover, our uniform algebraic definition of Wn,k gives us a uniform algebraic description of Permk(W).

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Outline

1

Background

2

The k-Parabolic Arrangement

3

Cellular Model

4

Discrete Morse Theory

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Discrete Morse Theory Discrete Morse theory - combinatorial approach to replacing Permk(W) with a ’smaller’ complex PermM

k (W) ∼

= Permk(W). It was originally discovered by Robin Forman. One hopes that the homology of PermM

k (W) is easy to compute.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Acyclic Matchings

  • Gk(W)− the directed Hasse diagram for Permk(W), with edges u → x

if u covers x. Definition M - a matching of Gk(W). Reverse the edges of M to obtain GM

k (W).

M is an acyclic matching if GM

k (W) has no directed cycles.

Motivation - edges of M correspond to cellular collapses.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Main Theorem of Discrete Morse Theory Given Permk(W), M, let Ci = set of unmatched i-cells. Theorem (R. Forman, 1996) There exists a CW complex PermM

k (W), such that

Permk(W) ∼ = PermM

k (W).

Moreover, the i-cells of PermM

k (W) are in bijection with Ci.

Two applications of Discrete Morse Theory are found here @ FPSAC:

1

Eric Clark, Richard Ehrenborg - The Frobenius Complex

2

Bridget Eileen Tenner - Boolean Complexes and Boolean Numbers

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Main result Theorem (Severs, W, 2009) Let k ≥ 3.

1

∃ an acyclic matching M on Permk(W), given by an algorithm.

2

The number of unmatched i-cells, |Ci|, is 0 unless i = t(k − 2) for some integer 0 ≤ t ≤ n

k .

3

In PermM

k (W), ∂σ = 0 for all σ ∈ ∆c.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Main result We can now conclude results concerning the homology of M(Wn,k). Theorem (Severs, W, 2009)

1

Hi(M(Wn,k)) ∼ = Hi(PermM

k (W))

2

Homology groups of M(Wn,k) are torsion-free.

3

Hi = 0 unless i = t(k − 2) for some 0 ≤ t ≤ n

k .

4

dim Hi(M(Wn,k)) = |Ci|.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Setup for Algorithm Given coset wWI, first consider the Dynkin diagram. Consider s2s6W{s3,s7} in W = E7, k = 3.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Setup for Algorithm Example: s2s6W{s3,s7} in W = E7, k = 3. Place a linear order on reflections.

1 2 3 4 5 6 7

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Setup for Algorithm Example: s2s6W{s3,s7} in W = E7, k = 3. Circle vertices corresponding to s ∈ I.

1 2 3 4 5 6 7

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Setup for Algorithm Example: s2s6W{s3,s7} in W = E7, k = 3. Write down minimal length element of wWI.

1 2 3 4 5 6 7

s2s6

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Setup for Algorithm Example: s2s6W{s3,s7} in W = E7, k = 3. Color descents of minimal element red.

1 2 3 4 5 6 7

s2s6

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Setup for Algorithm Example: s2s6W{s3,s7} in W = E7, k = 3. Goal: Attempt to circle/uncircle one black vertex.

1 2 3 4 5 6 7

s2s6

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Setup for Algorithm Example: s2s6W{s3,s7} in W = E7, k = 3. Cannot circle first vertex - eliminate s1, s3.

1 2 3 4 5 6 7

s2s6

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Setup for Algorithm Example: s2s6W{s3,s7} in W = E7, k = 3. We cannot circle s2, s4.

1 2 3 4 5 6 7

s2s6

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Setup for Algorithm Example: s2s6W{s3,s7} in W = E7, k = 3. We can circle s5.

1 2 3 4 5 6 7

s2s6

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Setup for Algorithm Example: s2s6W{s3,s7} in W = E7, k = 3. Match s2s6Ws3,s7 with s2s6Ws3,s5,s7.

1 2 3 4 5 6 7

s2s6

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Unmatched Elements Unmatched elements wWI correspond to cosets like the following:

1 2 3 4 5 6 7 8

s5

In this case, W = A8, k = 3, and the coset is s5Ws2,s3,s7,s8.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Unmatched Elements For wWI unmatched (w of minimal length), we have:

1

I consists of connected components of size k − 2.

2

each component has a black back neighbor.

3

vertices nonadjacent to any component must be red.

4

si is a descent of w ↔ si is red.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Example of B3,3 Algorithm can be used to find Betti numbers (W = B3, k = 3) Here we have the faces. dim # faces 48 1 72 2 12

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Example of B3,3 Algorithm can be used to find Betti numbers (W = B3, k = 3) For 0-faces, circle first black vertex. Thus, all but one of the 0-faces are matched. dim # faces 48 − 47 = 1 1 72 − 47 = 25 2 12

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Example of B3,3 Algorithm can be used to find Betti numbers (W = B3, k = 3) For 2-faces, uncircle first circled vertex in diagram. Thus, all 2-faces are matched. dim # faces 1 1 25 − 12 = 13 2 12 − 12 = 0

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Example of B3,3 Algorithm can be used to find Betti numbers (W = B3, k = 3) By our Theorem, we obtain the Betti numbers. i βi(M(B3,3)) 1 1 13 2

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Unmatched elements for B3

−1−2−3 123 −213 −2−13 −123 12−3 132 13−2 −3−12 −3−2−1 −3−12 −312 23−1 231

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Other Results One can compute the Betti numbers of exceptional types, and obtain (nasty) formulas for the infinite families. Approach of discrete Morse theory and cellular models work for other arrangements as well.

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Future Work Can these unmatched elements Ci be used to understand the action of W on Hi(M(Wn,k))?

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Background The k-Parabolic Arrangement Cellular Model Discrete Morse Theory

Questions? Thank you.

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