Local formality of inversion hyperplane arrangements William - PowerPoint PPT Presentation

Local formality of inversion hyperplane arrangements William Slofstra IQC, University of Waterloo July 15, 2016 joint work with Travis Scrimshaw Local formality of inversion hyperplane arrangements William Slofstra

Basic ideas Coxeter groups (combinatorics) Hyperplane arrangements Schubert varieties (geometry) (combinatorics) Local formality of inversion hyperplane arrangements William Slofstra

Inversion hyperplane arrangements Hyperplane arrangement: collection of subspaces of R n . W : finite Weyl group, such as S n Build hyperplane arrangement from inversions of w ∈ W . R = R + ∪ R − : root system of W . Inversions of w : α ∈ R + such that w − 1 α ∈ R − . Inversion hyperplane arrangement: � I ( w ) = ker α. inversions α Local formality of inversion hyperplane arrangements William Slofstra

Inversion hyperplane arrangements continued Inversions of w : α ∈ R + such that w − 1 α ∈ R − . Inversion hyperplane arrangement: � I ( w ) = ker α. inversions α Example ( W = S 4 ) R + = { e i − e j : 1 ≤ i < j ≤ 4 } e i − e j , i < j is an inversion if and only if w ( i ) > w ( j ) ker( e i − e j ) = { x ∈ C 4 : x i = x j } . I (3412) = ( x 2 = x 3 ) ∪ ( x 1 = x 3 ) ∪ ( x 2 = x 4 ) ∪ ( x 1 = x 4 ) ⊂ C 4 Local formality of inversion hyperplane arrangements William Slofstra

Schubert varieties versus hyperplane arrangements A hyperplane arrangement is free if its module of derivations is free. Theorem (S 2015) Schubert variety X ( w ) is rationally smooth if and only if I ( w ) is free and # of chambers = size of Bruhat interval How to see the connection? rootsystem pattern avoidance Local formality of inversion hyperplane arrangements William Slofstra

Root system pattern avoidance Rootsystem R lives in vector space V Root subsystem: R 0 = R ∩ V 0 for V 0 ⊆ V Weyl group W ( R 0 ) is a subgroup of W I ( w ) ∩ V 0 is the inversion set of an element w 0 ∈ W ( R 0 ), so get a flattening map fl V 0 : W → W ( R 0 ) A pattern is a pair ( w 1 , R 1 ) with w 1 ∈ W ( R 1 ) Defn (Billey-Postnikov): w ∈ W ( R ) contains ( w 1 , R 1 ) if • R 1 is isomorphic to a subsystem of R • fl R 0 ( w ) = w 1 Generalizes permutation pattern avoidance Local formality of inversion hyperplane arrangements William Slofstra

Root system pattern avoidance continued Defn (Billey-Postnikov): w ∈ W ( R ) contains ( w 1 , R 1 ) if • R 1 is isomorphic to a subsystem of R • fl R 0 ( w ) = w 1 Generalizes permutation pattern avoidance Theorem (Lakshmibai-Sandya) For W = S n , X ( w ) is smooth if and only if w avoids 3412 and 4231 Theorem (Billey, Billey-Postnikov) X ( w ) is (rationally) smooth if and only if w avoids a finite list of root system patterns. All patterns belong to stellar root systems ... only need three patterns to cover A , D , and E Local formality of inversion hyperplane arrangements William Slofstra

Connection with hyperplane arrangements Given: A = � α ∈ S ker α A flat of A is an intersection X of hyperplanes of A Flats of A correspond to linearly-closed subsets S 0 = S ∩ V 0 of S Localization of A is the arrangement � � A X = H = ker α X ⊆ H ∈A α ∈ S 0 A X has rank codim X = dim span S 0 . If A = I ( w ), then I ( w ) X = I (fl V 0 ( w )) Pattern avoidance criteria = check on localizations Local formality of inversion hyperplane arrangements William Slofstra

Pattern avoidance for freeness Theorem (S 2015) Schubert variety X ( w ) is rationally smooth if and only if I ( w ) is free and # of chambers = size of Bruhat interval Theorem (S 2016) An inversion arrangement I ( w ) is free if and only if w avoids a finite list of root system patterns. Proof uses concept from geometry of Schubert varieties Peterson translation Freeness is a local property: A free implies A X free Question: can we do this for other local properties of I ( w )? Local formality of inversion hyperplane arrangements William Slofstra

Terao’s conjecture Theorem (S 2016) An inversion arrangement I ( w ) is free if and only if w avoids a finite list of root system patterns. Conjecture (Terao) If matroid ( A ) ∼ = matroid ( B ) and A is free then B is free. Weak versions of conjecture: check for A and B in some family of arrangements Scrimshaw-S.: weak Terao’s conjecture true for inversion hyperplane arrangements. Local formality of inversion hyperplane arrangements William Slofstra

Terao’s conjecture and formality Terao’s conjecture has lead to the study of many other properties (combinatorial or not) which are “close” to freeness An arrangement is k -generated if cycle space spanned by k -cycles Formal if 3-generated Freeness implies local formality: A X formal for all flats X Local formality of inversion hyperplane arrangements William Slofstra

k -generatedness of inversion sets Theorem (Scrimshaw-S) Let σ ( R ) = min { k : for all w ∈ W and s ∈ span Inv ( w ) there exists X ⊂ I ( w ) with s ∈ span X and | X | ≤ k } Then min { k : I ( w ) is k-generated for all w ∈ W } = σ ( R ) + 1 and σ ( A n ) = σ ( B n ) = σ ( C n ) = σ ( F 4 ) = 3 , σ ( D n ) = σ ( E n ) = 4 Corollary (Scrimshaw-S) I ( w ) is locally formal if and only if w avoids a finite list of root system patterns. Consequently, local formality is a combinatorial property of inversion arrangements. Local formality of inversion hyperplane arrangements William Slofstra

k -generatedness of inversion sets ct’d Theorem (Scrimshaw-S) Let σ ( R ) = min { k : for all w ∈ W and s ∈ span Inv ( w ) there exists X ⊂ I ( w ) with s ∈ span X and | X | ≤ k } Then min { k : I ( w ) is k-generated for all w ∈ W } = σ ( R ) + 1 and σ ( A n ) = σ ( B n ) = σ ( C n ) = σ ( F 4 ) = 3 , σ ( D n ) = σ ( E n ) = 4 Proof uses reduced expressions... end up reducing to elements like s 1 s 3 s 2 in A 3 and s 1 s 2 s 3 s 0 in D 4 Stellar root systems arise “naturally”! Local formality of inversion hyperplane arrangements William Slofstra

Work in progress and open questions • Is there a combinatorial proof of Billey-Postnikov pattern avoidance criterion for smoothness in types ADE using bound on σ ( S )? (Conjecture: yes) • What about other properties? For instance, is there a bound on chordality for inversion sets? (Conjecture: no) Thanks! Local formality of inversion hyperplane arrangements William Slofstra