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
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory Outline Background 1 The k -Parabolic Arrangement 2 3 Cellular Model Discrete Morse Theory 4 k -Parabolic Arrangement C. Severs, J. White 2 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory Subspace arrangements A - is a collection of linear subspaces (over R n ). Assume ∩ X ∈ A X = 0. The complement M ( A ) = R n − ∪ 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¨ orner, Lov` asz, Yao k -Parabolic Arrangement C. Severs, J. White 3 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory The k -equal arrangement Definition The k -equal arrangment, A n − 1 , k consists of subspaces (of R n ): 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 hyperplane arrangement. k -Parabolic Arrangement C. Severs, J. White 4 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory The k -equal arrangement Definition The k -equal arrangment, A n − 1 , k consists of subspaces (of R n ): 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 hyperplane arrangement. In 1995, Bj¨ orner and Welker computed the homology of M ( A n , k ) . Homology groups are torsion free. H i is trivial unless i = t ( k − 2 ) , some 0 ≤ t ≤ n k . k -Parabolic Arrangement C. Severs, J. White 4 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory A type B and D k -equal arrangement In 1996, Bj¨ orner and Sagan defined type B and D analogues. Definition The D n , k arrangement consists of subspaces (of R n ): ± x i 1 = ± x i 2 = ... = ± x i k , 1 ≤ i 1 < · · · < i k ≤ n The B n , k , h arrangement consists of D n , k plus subspaces: x i 1 = . . . = x i h = 0, 1 ≤ i 1 < . . . < i h ≤ n They computed the homology of B n , k , h . In 2000, Kozlov-Feichtner computed the homology of D n , k , k > n 2 . Homology groups are torsion free k -Parabolic Arrangement C. Severs, J. White 5 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory The usual approach Usual approach to finding homology of M ( A ) : Apply Goresky MacPherson formula (to reduce problem to 1 studying homology of ∆( L ( A )) Show that ∆( L ( A )) is shellable. 2 Deduce that H i (∆( L ( A ))) is torsion-free, get Betti numbers. 3 Obtain results for M ( A ) . 4 k -Parabolic Arrangement C. Severs, J. White 6 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory The usual approach Usual approach to finding homology of M ( A ) : Apply Goresky MacPherson formula (to reduce problem to 1 studying homology of ∆( L ( A )) Show that ∆( L ( A )) is shellable. 2 Deduce that H i (∆( L ( A ))) is torsion-free, get Betti numbers. 3 Obtain results for M ( A ) . 4 NOTE THAT WE ARE NOT USING THIS APPROACH! k -Parabolic Arrangement C. Severs, J. White 6 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory Primary goals Generalize k -equal arrangement to any finite reflection group W . 1 Construct a cellular model for the complement. 2 Apply discrete Morse theory to our cellular model. 3 Obtain results regarding homology and Betti numbers. 4 k -Parabolic Arrangement C. Severs, J. White 7 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory Outline Background 1 The k -Parabolic Arrangement 2 3 Cellular Model Discrete Morse Theory 4 k -Parabolic Arrangement C. Severs, J. White 8 / 32
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 R n . S = set of simple reflections associated to W . R = { wsw − 1 : w ∈ W , s ∈ S } is the set of all reflections k -Parabolic Arrangement C. Severs, J. White 9 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory Coxeter Arrangement for W Definition The Coxeter arrangement is given by hyperplanes H r = { x ∈ R n : rx = x } for each r ∈ R . k -Parabolic Arrangement C. Severs, J. White 10 / 32
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. k -Parabolic Arrangement C. Severs, J. White 11 / 32
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 ∈ R n : wx = x , ∀ w ∈ G } For subspace X ⊂ R n , let Gal ( X ) = { w ∈ W : wx = x , ∀ x ∈ X } k -Parabolic Arrangement C. Severs, J. White 11 / 32
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 A 3 = S 4 . k -Parabolic Arrangement C. Severs, J. White 12 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory Example of correspondence W = S 4 L ( S 4 ) P ( S 4 ) 1234 <12,23,34> <12,23> <12,34> 13<12,34>13 34<12,23>34 23<12,34>23 <23,34> 12<23,34>12 13/24 123/4 134/2 12/34 1/234 124/3 14/23 24<12>24 <34> 23<12>23 <23> 34<23>34 <12> 1/24/3 13/2/4 12/3/4 1/2/34 1/23/4 14/2/3 e 1/2/3/4 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 Arrangement C. Severs, J. White 13 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory Definition of the k -parabolic arrangement Definition P n , k ( W ) = the collection of all k -parabolic subgroups of W . The k -parabolic arrangement W n , k is the collection of subspaces Fix ( G ) , G ∈ P n , k ( W ) k -Parabolic Arrangement C. Severs, J. White 14 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory Examples of the k -parabolic arrangement We recover some of the previous examples. W A n B n D n ( k = 3 ) D n ( k > 3) W ( k = 2 ) H ( W ) W n , k A n , k B n , k , k − 1 D n , 3 B n , k , k − 1 ∴ The k -equal arrangement has been generalized. k -Parabolic Arrangement C. Severs, J. White 15 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory Outline Background 1 The k -Parabolic Arrangement 2 3 Cellular Model Discrete Morse Theory 4 k -Parabolic Arrangement C. Severs, J. White 16 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory The Coxeter Complex and Permutahedron H ( W ) ∩ S n − 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 uW I , ordered by inclusion. Note: W I = < I > , for I ⊂ S . k -Parabolic Arrangement C. Severs, J. White 17 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory Picture k -parabolic subgroups → all parabolic subgroups (under ⊆ ) W n , k H ( W ) ∩ S n − 1 ∩ S n − 1 { wW I ⊇ k -parabolic cosets } → all standard parabolic cosets (under ⊇ ) C k ( W ) C ( W ) dualize dualize { wW I ⊇ k -parabolic cosets } → all standard parabolic cosets (under ⊆ ) Perm k ( W ) Perm ( W ) ∴ M ( W n , k ) is equivalent to removing Perm k ( W ) from Perm ( W ) . k -Parabolic Arrangement C. Severs, J. White 18 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory The subcomplex Perm k ( W ) Perm k ( W ) - subcomplex of W -Permutahedron, minimal nonfaces correspond to cosets wW I , such that W I is k -parabolic. Proposition M ( W n , k ) is homotopy equivalent to Perm k ( W ) . Proposition uses techniques of Bj¨ orner and Ziegler. k -Parabolic Arrangement C. Severs, J. White 19 / 32
Background The k -Parabolic Arrangement Cellular Model Discrete Morse Theory Example with B 3 , 3 H ( B 3 ) B 3 -Permutahedron B 3 , 3 k -Parabolic Arrangement C. Severs, J. White 20 / 32
Recommend
More recommend
