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) Schubert varieties (geometry) Hyperplane arrangements (combinatorics)

Local formality of inversion hyperplane arrangements William Slofstra

Inversion hyperplane arrangements

Hyperplane arrangement: collection of subspaces of Rn. W : finite Weyl group, such as Sn Build hyperplane arrangement from inversions of w ∈ W . R = R+ ∪ R−: root system of W . Inversions of w: α ∈ R+ such that w−1α ∈ R−. Inversion hyperplane arrangement: I(w) =

• inversions α

ker α.

Local formality of inversion hyperplane arrangements William Slofstra

Inversion hyperplane arrangements continued

Inversions of w: α ∈ R+ such that w−1α ∈ R−. Inversion hyperplane arrangement: I(w) =

• inversions α

ker α.

Example (W = S4)

R+ = {ei − ej : 1 ≤ i < j ≤ 4} ei − ej, i < j is an inversion if and only if w(i) > w(j) ker(ei − ej) = {x ∈ C4 : xi = xj}. I(3412) = (x2 = x3) ∪ (x1 = x3) ∪ (x2 = x4) ∪ (x1 = x4) ⊂ C4

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: R0 = R ∩ V0 for V0 ⊆ V Weyl group W (R0) is a subgroup of W I(w) ∩ V0 is the inversion set of an element w0 ∈ W (R0), so get a flattening map flV0 : W → W (R0) A pattern is a pair (w1, R1) with w1 ∈ W (R1) Defn (Billey-Postnikov): w ∈ W (R) contains (w1, R1) if

• R1 is isomorphic to a subsystem of R
• flR0(w) = w1

Generalizes permutation pattern avoidance

Local formality of inversion hyperplane arrangements William Slofstra

Root system pattern avoidance continued

Defn (Billey-Postnikov): w ∈ W (R) contains (w1, R1) if

• R1 is isomorphic to a subsystem of R
• flR0(w) = w1

Generalizes permutation pattern avoidance

Theorem (Lakshmibai-Sandya)

For W = Sn, 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 S0 = S ∩ V0 of S Localization of A is the arrangement AX =

• X⊆H∈A

H =

• α∈S0

ker α AX has rank codimX = dim spanS0. If A = I(w), then I(w)X = I(flV0(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 AX 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: AX 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(An) = σ(Bn) = σ(Cn) = σ(F4) = 3, σ(Dn) = σ(En) = 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(An) = σ(Bn) = σ(Cn) = σ(F4) = 3, σ(Dn) = σ(En) = 4 Proof uses reduced expressions... end up reducing to elements like s1s3s2 in A3 and s1s2s3s0 in D4 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

• n σ(S)? (Conjecture: yes)
• What about other properties? For instance, is there a bound
• n chordality for inversion sets? (Conjecture: no)

Thanks!

Local formality of inversion hyperplane arrangements William Slofstra