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

Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane CNRS, IGM, Universit´ e Paris-Est ACPMS Online Seminar 26 june 2020 joint work with Matthieu Josuat-Verg` es Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

Noncrossing partitions and interval partitions Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

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. Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

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. • • • • • • • • • • j i k l Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

Example { 1 , 4 , 5 } ∪ { 2 } ∪ { 3 } ∪ { 6 , 8 } ∪ { 7 } can be drawn on a circle without crossings 1 2 8 3 7 6 4 5 Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

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

Non-crossing cumulants (Speicher 1993) ( A , τ )=an algebra with a linear form. One defines multilinear functionals R n by � τ ( a 1 . . . a n ) = R π ( a 1 , . . . , a n ) π ∈ NC ( n ) � R π ( a 1 , . . . , a n ) = R | p | ( a i 1 , . . . , a i | p | ) part of π where p = { i 1 , . . . , i | p | } is a part of π . Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

Examples: τ ( a 1 ) = R 1 ( a 1 ) { 1 } τ ( a 1 a 2 ) = R 2 ( a 1 , a 2 ) { 1 , 2 } + R 1 ( a 1 ) R 1 ( a 2 ) { 1 } ∪ { 2 } so that R 1 ( a ) = τ ( a ) R 2 ( a 1 , a 2 ) = τ ( a 1 a 2 ) − τ ( a 1 ) τ ( a 2 ) Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

τ ( a 1 a 2 a 3 ) = R 3 ( a 1 , a 2 , a 3 ) { 1 , 2 , 3 } + R 1 ( a 1 ) R 2 ( a 2 , a 3 ) { 1 } ∪ { 2 , 3 } + R 2 ( a 1 , a 3 ) R 1 ( a 2 ) { 1 , 3 } ∪ { 2 } + R 2 ( a 1 , a 2 ) R 1 ( a 3 ) { 1 , 2 } ∪ { 3 } + R 1 ( a 1 ) R 1 ( a 2 ) R 1 ( a 3 ) { 1 } ∪ { 2 } ∪ { 2 } R 3 ( a 1 , a 2 , a 3 ) = τ ( a 1 a 2 a 3 ) − τ ( a 1 a 2 ) τ ( a 3 ) − τ ( a 1 a 3 ) τ ( a 2 ) − τ ( a 1 ) τ ( a 2 a 3 ) + 2 τ ( a 1 ) τ ( a 2 ) τ ( a 3 ) Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

In general there is an inversion formula: � R n ( a 1 , . . . , a n ) = µ ( π ) τ π ( a 1 , . . . , a n ) π ∈ NC ( n ) where µ is a M¨ obius function on NC ( n ). Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

In general there is an inversion formula: � R n ( a 1 , . . . , a n ) = µ ( π ) τ π ( a 1 , . . . , a n ) π ∈ NC ( n ) where µ is a M¨ obius function on NC ( n ). The vanishing of noncrossing cumulants allow to define the notion of free independence (due to Voiculescu). Remark: if one has a commutative algebra and one uses the lattice of all set partitions, one obtains Rota’s approach to independence in classical probability. Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

A set partition of { 1 , . . . , n } is an interval partition if its parts are intervals. Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

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 2 n − 1 for the inverse refinement order, a sublattice of NC ( n ). Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

One can define Boolean cumulants (Speicher, Woroudi 1997) by � τ ( a 1 . . . a n ) = B π ( a 1 , . . . , a n ) π ∈ I ( n ) � B π ( a 1 , . . . , a n ) = B | p | ( a i 1 , . . . , a i | p | ) part of π where p = { i 1 , . . . , i | p | } is a part of π . Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

One can define Boolean cumulants (Speicher, Woroudi 1997) by � τ ( a 1 . . . a n ) = B π ( a 1 , . . . , a n ) π ∈ I ( n ) � B π ( a 1 , . . . , a n ) = B | p | ( a i 1 , . . . , a i | p | ) part of π where p = { i 1 , . . . , i | p | } is a part of π . Again there is an inversion formula: � B n ( a 1 , . . . , a n ) = µ ( π ) τ π ( a 1 , . . . , a n ) π ∈ I ( n ) where µ is a M¨ obius function on I ( n ). Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

One can define Boolean cumulants (Speicher, Woroudi 1997) by � τ ( a 1 . . . a n ) = B π ( a 1 , . . . , a n ) π ∈ I ( n ) � B π ( a 1 , . . . , a n ) = B | p | ( a i 1 , . . . , a i | p | ) part of π where p = { i 1 , . . . , i | p | } is a part of π . Again there is an inversion formula: � B n ( a 1 , . . . , a n ) = µ ( π ) τ π ( a 1 , . . . , a n ) π ∈ I ( n ) where µ is a M¨ obius function on I ( n ). Boolean cumulants allow to define the notion of Boolean independence . Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

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. Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

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. Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

Noncrossing partitions and interval partitions I ( n ) ⊂ NC ( n ) as a Boolean sublattice. Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

One can embed noncrossing partitions inside the symmetric group: 1 2 8 3 7 6 4 5 (145)(68) ∈ S 8 The parts of the partition are the cycles of a permutation. Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

(123) (12) (13) (23) e I (3) ⊂ NC (3)( ⊂ S 3 ) Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

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

Noncrossing and interval partitions and orders on S n 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 } Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

Finite Coxeter groups. V =finite dimensional euclidean space. O ( V )=orthogonal group. Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

Finite Coxeter groups. V =finite dimensional euclidean space. O ( V )=orthogonal group. W = finite subgroup of O ( V ) generated by orthogonal reflections. Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

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, v � α ± α ∈ R R is the set of roots of W . Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

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, v � α ± 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 Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

Example: the symmetric group S n acts on V = { ( x 1 , . . . , x n ) | � i x i = 0 } by permutation of coordinates. T = { ( ij ); i < j } S = { ( i i + 1); i < n } Π = { e i − e j ; i < j } ∆ = { e i − e i +1 , i < n } Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

S 3 : fundamental cone: { x 1 > x 2 > x 3 ; x 1 + x 2 + x 3 = 0 } reflection hyperplanes: x 1 = x 2 ; x 1 = x 3 ; x 2 = x 3 ; the root system: negative roots, simple roots Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane

A Coxeter element is a product of the simple generators of W in some order. c = s i 1 s i 2 . . . s i n Noncrossing partitions, interval partitions and the Bruhat order Philippe Biane