Noncrossing partitions, interval partitions and the Bruhat order - - PowerPoint PPT Presentation

Noncrossing partitions, interval partitions and the Bruhat

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Philippe Biane CNRS, IGM, Universit´ e Paris-Est ACPMS Online Seminar 26 june 2020 joint work with Matthieu Josuat-Verg` es

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Noncrossing partitions and interval partitions

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

A set partition of {1, . . . , n} is non-crossing if there are no (i, j, k, l) with i < j < k < l and i ∼ k, j ∼ l and i, j not in same part.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

A set partition of {1, . . . , n} is non-crossing if there are no (i, j, k, l) with i < j < k < l and i ∼ k, j ∼ l and i, j not in same part.

• i
• j
• k
• l
• Philippe Biane

Noncrossing partitions, interval partitions and the Bruhat order

Example {1, 4, 5} ∪ {2} ∪ {3} ∪ {6, 8} ∪ {7} can be drawn on a circle without crossings

1 2 3 4 5 6 7 8

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Basic facts NC(n) is a lattice for the inverse refinement order. |NC(n)| = Cat(n) = (2n)! n!(n + 1)! Many papers on this structure since the first one by Kreweras in 1972.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Non-crossing cumulants (Speicher 1993) (A, τ)=an algebra with a linear form. One defines multilinear functionals Rn by τ(a1 . . . an) =

• π∈NC(n)

Rπ(a1, . . . , an) Rπ(a1, . . . , an) =

• part ofπ

R|p|(ai1, . . . , ai|p|) where p = {i1, . . . , i|p|} is a part of π.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Examples: τ(a1) = R1(a1) {1} τ(a1a2) = R2(a1, a2) {1, 2} +R1(a1)R1(a2) {1} ∪ {2} so that R1(a) = τ(a) R2(a1, a2) = τ(a1a2) − τ(a1)τ(a2)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

τ(a1a2a3) = R3(a1, a2, a3) {1, 2, 3} +R1(a1)R2(a2, a3) {1} ∪ {2, 3} +R2(a1, a3)R1(a2) {1, 3} ∪ {2} +R2(a1, a2)R1(a3) {1, 2} ∪ {3} +R1(a1)R1(a2)R1(a3) {1} ∪ {2} ∪ {2} R3(a1, a2, a3) = τ(a1a2a3) − τ(a1a2)τ(a3) − τ(a1a3)τ(a2) −τ(a1)τ(a2a3) + 2τ(a1)τ(a2)τ(a3)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

In general there is an inversion formula: Rn(a1, . . . , an) =

• π∈NC(n)

µ(π)τπ(a1, . . . , an) where µ is a M¨

• bius function on NC(n).

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

In general there is an inversion formula: Rn(a1, . . . , an) =

• π∈NC(n)

µ(π)τπ(a1, . . . , an) where µ is a M¨

• bius function on NC(n).

The vanishing of noncrossing cumulants allow to define the notion

• f free independence (due to Voiculescu).

Remark: if one has a commutative algebra and one uses the lattice

• f all set partitions, one obtains Rota’s approach to independence

in classical probability.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

A set partition of {1, . . . , n} is an interval partition if its parts are intervals.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

A set partition of {1, . . . , n} is an interval partition if its parts are intervals. An interval partition is determined by the first member of each of its parts. It follows that the set of intervals partitions I(n) is a Boolean lattice of order 2n−1 for the inverse refinement order, a sublattice

• f NC(n).

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

One can define Boolean cumulants (Speicher, Woroudi 1997) by τ(a1 . . . an) =

• π∈I(n)

Bπ(a1, . . . , an) Bπ(a1, . . . , an) =

• part ofπ

B|p|(ai1, . . . , ai|p|) where p = {i1, . . . , i|p|} is a part of π.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

One can define Boolean cumulants (Speicher, Woroudi 1997) by τ(a1 . . . an) =

• π∈I(n)

Bπ(a1, . . . , an) Bπ(a1, . . . , an) =

• part ofπ

B|p|(ai1, . . . , ai|p|) where p = {i1, . . . , i|p|} is a part of π. Again there is an inversion formula: Bn(a1, . . . , an) =

• π∈I(n)

µ(π)τπ(a1, . . . , an) where µ is a M¨

• bius function on I(n).

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

One can define Boolean cumulants (Speicher, Woroudi 1997) by τ(a1 . . . an) =

• π∈I(n)

Bπ(a1, . . . , an) Bπ(a1, . . . , an) =

• part ofπ

B|p|(ai1, . . . , ai|p|) where p = {i1, . . . , i|p|} is a part of π. Again there is an inversion formula: Bn(a1, . . . , an) =

• π∈I(n)

µ(π)τπ(a1, . . . , an) where µ is a M¨

• bius function on I(n).

Boolean cumulants allow to define the notion of Boolean independence.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Speicher has shown that classical independence, free independence and Boolean independence are the only general notions of “probabilistic independence” in algebras, satisfying some natural requirements.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Speicher has shown that classical independence, free independence and Boolean independence are the only general notions of “probabilistic independence” in algebras, satisfying some natural requirements. In this talk I will show how these combinatorial structures

• noncrossing and interval partitions- are deeply related to the

theory of Coxeter groups.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Noncrossing partitions and interval partitions I(n) ⊂ NC(n) as a Boolean sublattice.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

One can embed noncrossing partitions inside the symmetric group:

1 2 3 4 5 6 7 8

(145)(68) ∈ S8 The parts of the partition are the cycles of a permutation.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

(12) (123) (23) e (13) I(3) ⊂ NC(3)(⊂ S3)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

(1234) () (123) (12) (12)(34) (23) (234) (34) (124) (24) (134) (13) (14)(23) (14)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Noncrossing and interval partitions and orders on Sn Let c = (123 . . . n): Noncrossing partitions are the permutations σ such that σ ≤T c for the absolute order ≤T. NC(n) = {σ|σ ≤T c} Interval partitions are the permutations σ such that σ ≤B c for the Bruhat (or strong) order ≤B. I(n) = {σ|σ ≤B c}

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Finite Coxeter groups. V =finite dimensional euclidean space. O(V )=orthogonal group.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Finite Coxeter groups. V =finite dimensional euclidean space. O(V )=orthogonal group. W = finite subgroup of O(V ) generated by orthogonal reflections.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Finite Coxeter groups. V =finite dimensional euclidean space. O(V )=orthogonal group. W = finite subgroup of O(V ) generated by orthogonal reflections. T ⊂ W subset of reflections in W , t ∈ T given by t(v) = v − α, vα ± α ∈ R R is the set of roots of W .

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Finite Coxeter groups. V =finite dimensional euclidean space. O(V )=orthogonal group. W = finite subgroup of O(V ) generated by orthogonal reflections. T ⊂ W subset of reflections in W , t ∈ T given by t(v) = v − α, vα ± α ∈ R R is the set of roots of W . S ⊂ T=reflections with respect to the hyperplanes bounding a fundamental domain of W . =simple system of generators R = Π ∪ (−Π) (Π= positive roots) ∆ ⊂ Π: roots of simple reflections

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Example: the symmetric group Sn acts on V = {(x1, . . . , xn)|

i xi = 0} by permutation of

coordinates. T = {(ij); i < j} S = {(i i + 1); i < n} Π = {ei − ej; i < j} ∆ = {ei − ei+1, i < n}

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

S3: fundamental cone: {x1 > x2 > x3; x1 + x2 + x3 = 0} reflection hyperplanes: x1 = x2; x1 = x3; x2 = x3; the root system: negative roots, simple roots

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

A Coxeter element is a product of the simple generators of W in some order. c = si1si2 . . . sin

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

A Coxeter element is a product of the simple generators of W in some order. c = si1si2 . . . sin They are all conjugate in W .

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

A Coxeter element is a product of the simple generators of W in some order. c = si1si2 . . . sin They are all conjugate in W . In Sn the Coxeter elements are cycles of maximal length.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Absolute length T is a generating system of W . ℓT= the associated length function on W

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Absolute length T is a generating system of W . ℓT= the associated length function on W ℓT(v)=minimal number of reflections in a decomposition v = t1 . . . tk One has ℓT(v) = codim(Fix(v)) For the symmetric group ℓT(σ) = n − ♯(cycles of σ)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Absolute order If vw−1 ∈ T then v <T w iff ℓT(v) < ℓT(w) By transitivity this defines the absolute order <T on W .

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Absolute order If vw−1 ∈ T then v <T w iff ℓT(v) < ℓT(w) By transitivity this defines the absolute order <T on W . The order can also be characterized using the triangle inequality: v ≤T w ⇐ ⇒ ℓT(v) + ℓT(v−1w) = ℓT(w) v is on a geodesic from e to w in the Cayley graph of W .

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Bruhat length S is a generating system of W ℓB= the associated length function on W

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Bruhat length S is a generating system of W ℓB= the associated length function on W ℓB(v)=minimal number of reflections in a decomposition into simple reflections v = s1 . . . sk One has ℓB(v) = ♯right inversions of v = ♯{t ∈ T|ℓB(vt) < ℓB(v)} For the symmetric group ℓB(σ) = ♯{i < j|σ(i) > σ(j)}

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Bruhat order If vw−1 ∈ T then v <B w iff ℓB(v) < ℓB(w) By transitivity one obtains the Bruhat order on W . One has v ≤B w if and only if v has a reduced decomposition which is a subword of a reduced decomposition of w. w = s1 . . . sk si ∈ S v = si1 . . . sij 1 ≤ i1 < i2 < . . . < ij ≤ k

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Non-crossing and interval partitions c= Coxeter element NC(W , c) = {v ∈ W |v ≤T c} NC(W , c) is a lattice for ≤T (Bessis, Brady and Watt)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Non-crossing and interval partitions c= Coxeter element NC(W , c) = {v ∈ W |v ≤T c} NC(W , c) is a lattice for ≤T (Bessis, Brady and Watt) Int(W , c) = {v ∈ W |v ≤B c} Int(W , c) ⊂ NC(W , c) Int(W , c) is a Boolean lattice for ≤B (or ≤T) with 2ℓB(c) = 2|S| elements If c = s1 . . . sn then v ∈ Int(W , c) = si1 . . . sik i1 < i2 < . . . < ik

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

(1234) () (123) (12) (12)(34) (23) (234) (34) (124) (24) (134) (13) (14)(23) (14)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

The orders ⊏ and ≪ If v ⋖T w then vw−1 ∈ T therefore either v <B w or v >B w. We put v ⊏ · w if v ⋖T w and v <B w v ≪ · w if v ⋖T w and v >B w By transitivity one gets two orders ⊏ and ≪ on W . We study the restriction of these orders to NC(W , c).

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

The orders ⊏ and ≪ on NC3 (12) (123) (23) e (13)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

(1234) () (123) (12) (12)(34) (23) (234) (34) (124) (24) (134) (13) (14)(23) (14)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

(1243) () (123) (12) (12)(34) (23) (243) (34) (124) (24) (143) (13) (13)(24) (14)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Parabolic subgroups: for v ∈ W Γ(w) = {v ∈ W |Fix(w) ⊂ Fix(v)} is a reflection group R(Γ(w)) ⊂ R(W ) and the decomposition R = Π ∪ (−Π) allows to define positive roots in Γ(w), as well as generators and simple roots ∆(Γ(w)). If W = Sn, then Γ(σ) is the subgroup preserving the cycles of σ.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Kreweras complement If w ∈ NC(W , c) then K(w) = w−1c ∈ NC(W , c) K : NC(W , c) → NC(W , c) is the Kreweras complement. wK(w) = c K reverses absolute order: v ≤T w ⇐ ⇒ K(w) ≤T K(v)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Proposition

• 1. If v, w ∈ NC(W , c) and v ⋖T w, then

v >B w = ⇒ K(v) >B K(w).

• 2. If v, w ∈ NC(W , c), v ⋖T w and v has full support in

NC(W , c) (every reduced decomposition of v uses all the simple reflections) then K(v) >B K(w) = ⇒ v >B w.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Characterization of ⊏ If v ⊏ w then v ≤T w and v ≤B w but the converse is not true. Proposition Let v, w ∈ NC(W , c) then v ⊏ w ⇐ ⇒ ∆(Γ(v)) ⊂ ∆(Γ(w)) Reminder: v ≤T w ⇐ ⇒ Γ(v) ⊂ Γ(w) Corollary The order ideals Iv = {w ∈ NC(W , c)|w ⊏ v} are Boolean lattices.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Caracterization of ≪ If v ≪ w then v ≤T w and w ≤B v but the converse is not true. Proposition Let v, w ∈ NC(W , c) be such that v ≤T w then v ≪ w ⇐ ⇒ v has full support in Γ(w) Let Jw = {v ∈ NC(W , c)|w ≪ v} then K(Jw) = IK(w) The order ideals Jw are Boolean lattices.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Intervals for ⊏ There is a unique interval of maximal length: [e, c] Two interval types in length n − 1: [e, K(t)] t ∈ T [s, c] s ∈ S These intervals are in bijection with (−∆) ∪ Π the set of almost positive roots. This suggests connections with cluster algebras.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

(1234) () (123) (12) (12)(34) (23) (234) (34) (124) (24) (134) (13) (14)(23) (14)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

The cluster complex (Fomin and Zelevinsky) One defines a binary symmetric relation c on (−∆) ∪ Π (almost positive roots):

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

The cluster complex (Fomin and Zelevinsky) One defines a binary symmetric relation c on (−∆) ∪ Π (almost positive roots): ◮ If α, β ∈ −∆ then α c β ◮ If α ∈ −∆ and β ∈ Π then α c β if and only if r(−α) is not in the support of r(β). ◮ If α, β ∈ Π then α c β if and only if (r(α)r(β) ≤T c or r(β)r(α) ≤T c) and α, β ≥ 0.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

The cluster complex (Fomin and Zelevinsky) One defines a binary symmetric relation c on (−∆) ∪ Π (almost positive roots): ◮ If α, β ∈ −∆ then α c β ◮ If α ∈ −∆ and β ∈ Π then α c β if and only if r(−α) is not in the support of r(β). ◮ If α, β ∈ Π then α c β if and only if (r(α)r(β) ≤T c or r(β)r(α) ≤T c) and α, β ≥ 0. The cluster complex is the simplicial complex made of subsets X ⊂ (−∆) ∪ Π such that x1, x2 ∈ X = ⇒ x1 c x2. It is dual to the associahedron.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Proposition The number of intervals of height k for ⊏ is equal to the number of faces of cardinal n − k of the cluster complex (and to the number of faces of cardinal k of the associahedron). There exists a bijective proof when c is bipartite.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Intervals for ≪ Proposition The number of intervals [v, w]≪ with rk(w) = k is equal to the number of faces of cardinal k of the positive cluster complex (the restriction of the cluster complex to positive roots) . There exists a bijection in the general case.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

(NC(W , c), ⊏) as a simplicial complex Define the binary relation ≬c on Π: α ≬c β if and only if : r(α)r(β) ≤T c or r(β)r(α) ≤T c and α, β ≤ 0. Ξ(W , c): is the associated simplicial complex.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

(NC(W , c), ⊏) as a simplicial complex Define the binary relation ≬c on Π: α ≬c β if and only if : r(α)r(β) ≤T c or r(β)r(α) ≤T c and α, β ≤ 0. Ξ(W , c): is the associated simplicial complex. Remark: r(α)r(β) ≤T c or r(β)r(α) ≤T c is the general Coxeter analogue of the non-crossing relation.

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

(NC(W , c), ⊏) as a simplicial complex Define the binary relation ≬c on Π: α ≬c β if and only if : r(α)r(β) ≤T c or r(β)r(α) ≤T c and α, β ≤ 0. Ξ(W , c): is the associated simplicial complex. Remark: r(α)r(β) ≤T c or r(β)r(α) ≤T c is the general Coxeter analogue of the non-crossing relation. Proposition: w → ∆(Γ(w)) is a bijection from NC(W , c) onto Ξ(W , c).

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

(12) (23) (13) Graph of the relation ≬c (12) (123) (23) e (13)

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

(12) (34) (23) (14) (24) (13) (1234) () (123) (12) (12)(34) (23) (234) (34) (124) (24) (134) (13) (14)(23) (14) Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order

Thank you!

Philippe Biane Noncrossing partitions, interval partitions and the Bruhat order