k-Parabolic Subspace Arrangements H el` ene Barcelo Christopher - PowerPoint PPT Presentation

k-Parabolic Subspace Arrangements H´ el` ene Barcelo Christopher Severs Jacob White Mathematical Sciences Research Insitute and ASU School of Mathematics and Statistical Sciences

Introduction Fadell, Fox, Neuwirth, 1963 Take an n dimensional complex space, delete all diagonals z i = z j C n − D is a K ( π, 1 ) space, with fundamental group ∼ = pure braid group Khovanov, 1996 (real counterpart) Take an n dimensional real space, delete all co-dimension 2 subspaces, x i = x j = x k R n − X n is a K ( π, 1 ) space, with fundamental group ∼ = pure twin group k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 2 / 33

Introduction Brieskorn, Deligne 1970’s Consider H , the complexification of a Coxeter arrangement of type W C n − H is a K ( π, 1 ) space, with fundamental group ∼ = pure Artin group, of type W. B. Severs, White, 2008 (real counterpart) Take an n dimensional real space, delete P W , the set of all 3-parabolic subspaces of type W R n − P W is a K ( π, 1 ) space ( claim ) with fundamental group ∼ = pure triplet group of type W (Theorem) k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 3 / 33

Tools Discrete Homotopy Theory k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 4 / 33

Reflection group W - an irreducible finite real reflection group acting on R n , with: S ⊂ W a set of simple reflections, R = { wsw − 1 : s ∈ S , w ∈ W } the set of all reflections. m ( s , s ) = 1 , m ( s , t ) = m ( t , s ) for all s , t ∈ S and k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 5 / 33

A presntation of W W is generated by S subject to: s 2 = 1, ∀ s ∈ S 1 st = ts , ∀ s , t ∈ S such that m ( s , t ) = 2 2 sts = tst , ∀ s , t ∈ S , such that m ( s , t ) = 3 3 . . . i. stst · · · = tsts · · · , ∀ s , t ∈ S , such that m ( s , t ) = i � �� � � �� � i i . . . k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 6 / 33

Coxeter Arrangement for W Definition The Coxeter arrangement H ( W ) is given by hyperplanes H r = { x ∈ R n : rx = x } for each r ∈ R . k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 7 / 33

Example: Braid Arrangement When W is of type A , the Coxeter arrangement is given by x i − x j = 0 , 1 ≤ i < j ≤ n + 1 and π 1 ( C n − H A ) ∼ = pure braid group π 1 ( R n − H W ) ∼ = pure Artin group of type W . k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 8 / 33

What is an Artin group of type W over C W 1 is generated by S ∈ W subject to: s 2 = 1, ∀ s ∈ S 1 st = ts , ∀ s , t ∈ S such that m ( s , t ) = 2 2 sts = tst , ∀ s , t ∈ S , such that m ( s , t ) = 3 3 . . . i stst · · · = tsts · · · , ∀ s , t ∈ S , such that m ( s , t ) = i � �� � � �� � i i . . . k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 9 / 33

Pure Artin group of type W Let ϕ : W 1 → W with ϕ ( s ) = s , for all s ∈ S . ker ϕ = pure Artin Group of type W. Brieskorns 1973 H W a Coxeter arrangement of type W π 1 ( C n − H W ) ∼ = ker ϕ k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 10 / 33

The k -equal arrangement Definition The k -equal arrangment, A n , k consists of subspaces (of R n ) given by equations: x i 1 = x i 2 = ... = x i k , for all distinct indices 1 ≤ i 1 < ... < i k ≤ n When k=2 we recover the Braid arrangement. k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 11 / 33

The k -equal arrangement Definition The k -equal arrangment, A n , k consists of subspaces (of R n ) given by equations: x i 1 = x i 2 = ... = x i k , for all distinct indices 1 ≤ i 1 < ... < i k ≤ n When k=2 we recover the Braid arrangement. Khovanov (1996) gave a description of π 1 ( M ( A n , 3 )) as a Pure Twin Group. He also showed that M R ( A n , 3 ) is a K ( π, 1 ) space. k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 11 / 33

What is a twin group over R W 3 is generated by S ∈ S n subject to: s 2 = 1, ∀ s ∈ S 1 st = ts , ∀ s , t ∈ S such that m ( s , t ) = 2 2 sts = tst , ∀ s , t ∈ S , such that m ( s , t ) = 3 3 k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 12 / 33

Pure twin group Let ϕ : W 3 → W with ϕ ( s ) = s , for all s ∈ S . ker ϕ = pure Twin Group ∼ = π 1 ( M ( A n , 3 )) k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 13 / 33

Real analogue to Brieskorn results in the spirit of Khovanov W i is generated by S ⊂ W subject to: s 2 = 1, ∀ s ∈ S 1 st = ts , ∀ s , t ∈ S such that m ( s , t ) = 2 2 sts = tst , ∀ s , t ∈ S , such that m ( s , t ) = 3 3 . . . i. stst · · · = tsts · · · , ∀ s , t ∈ S , such that m ( s , t ) = i � �� � � �� � i i . . . k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 14 / 33

Example with Dynkin Diagrams Comparing Dynkin diagrams of W and W ′ : A ′ A 5 5 ∞ ∞ ∞ ∞ D ′ D 5 5 ∞ ∞ ∞ ∞ k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 15 / 33

k -Parabolic Subspace Arrangement Let ϕ : W i → W given by ϕ ( s ) = s , for all s ∈ S . Theorem B., Severs, White, (2008) ker ϕ ∼ = π 1 ( M R ( W n , 3 )) , where W n , k is the k-parabolic arrangement of type w. k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 16 / 33

Parabolic Subgroups and more Definition A subgroup G ⊆ W is parabolic if G = < wIw − 1 > , for some I ⊂ S , w ∈ W . G is k -parabolic if G is of rank k − 1. k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 17 / 33

Parabolic Subgroups and more Definition A subgroup G ⊆ W is parabolic if G = < wIw − 1 > , for some I ⊂ S , w ∈ W . G is k -parabolic if G is of rank k − 1. For G ⊂ W , let Fix ( G ) = { x ∈ R n : wx = x , ∀ w ∈ G } For subspace X ⊂ R n , let Gal ( X ) = { w ∈ W : wx = x , ∀ x ∈ X } k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 17 / 33

Galois Correspondence Let P ( W ) be the poset of all parabolic subgroups of W ordered by inclusion. Let L ( W ) be the intersection lattice of the Coxeter arrangement, ordered by reverse inclusion. Theorem (Barcelo and Ihrig, 1999) P ( W ) ∼ = L ( W ) via G → Fix ( G ) Gal ( X ) ← X We will use this “Galois correspondence” to define k -parabolic arrangements. But first we give an example for A 3 = S 4 . k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 18 / 33

Example of correspondence W = S 4 L ( S 4 ) P ( S 4 ) 1234 <12, 23, 34> 13<12,34>13 <12,34> 34<12,23>34 <12,23> 14/23 1/234 13/24 123/4 134/2 <23,34> 23<12,34>23 12<23,34>12 124/3 12/34 1/24/3 13/2/4 1/23/4 14/2/3 12/3/4 1/2/34 24<12>24 23<12>23 <34> 34<23>34 <12> <23> 1/2/3/4 e 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 ) k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 19 / 33

Definition of the k -parabolic arrangement Definition Let W be an irreducible real reflection group of rank n . Let P n , k ( W ) contain all irreducible k -parabolic subgroups of W . Then the k -parabolic arrangement W n , k is the collection of subspaces Fix ( G ) , G ∈ P n , k ( W ) k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 20 / 33

Example with W = A 8 Example: W = S 9 = A 8 Let G = ( 1 , 4 )( 6 , 8 ) < ( 4 , 5 ) , ( 5 , 6 ) > ( 6 , 8 )( 1 , 4 ) = < ( 1 , 4 ) , ( 4 , 8 ) > . We see that Fix ( G ) is given by x 1 = x 4 = x 8 . For every G ∈ P 8 , 3 ( A 8 ) , Fix ( G ) is a subspace in A 9 , 3 . Thus, W 8 , 3 is the 3-equal arrangement in R 8 . k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 21 / 33

Examples of the k -parabolic arrangement When W is of type A or B , then W n , k corresponds to A n + 1 , k , and the B n , k , k − 1 arrangement (of Bj¨ orner-Welker and Bj¨ orner-Sagan respectively). W n , 2 is the Coxeter arrangement for W and W n , n + 1 consists of the origin. When W is of type D , then W n , 3 corresponds to the Bj¨ orner-Sagan D n , 3 arrangement (not so for W n , k , k > 3). k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 22 / 33

Why is B., Severs and White’s Theorem true? Essentially because π 1 ( M R ( W n , 3 )) ∼ = A n − 2 ( W − permutahedron ) 1 and ker ( ϕ ) ∼ = A n − 2 ( W − permutahedron ) . 1 k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 23 / 33

What is A 1 Definition Let ∆ be simplicial complex of dimension d , 0 ≤ q ≤ d , σ 0 ∈ ∆ be maximal with dimension ≥ q . Two simplicies σ and τ are q -near if | σ ∩ τ | ≥ q + 1. 1 A q -chain is a sequence σ 1 , . . . , σ k , such that σ i , σ i + 1 are q -near for all i . 2 A q -loop is a q -chain with σ 1 = σ k = σ 0 . 3 k-Parabolic Subspace Arrangements H. Barcelo MATHEMATICS AND STATISTICS 24 / 33