Positive m -divisible non-crossing partitions and their cylic - PowerPoint PPT Presentation

Positive m -divisible non-crossing partitions and their cylic sieving Christian Krattenthaler and Stump Universit¨ at Wien and Freie Universit¨ at Berlin Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Let W be a finite real reflection group. The absolute length ( reflection length ) ℓ T ( w ) of an element w ∈ W is defined by the smallest k such that w = t 1 t 2 · · · t k , where all t i are reflections. Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Let W be a finite real reflection group. The absolute length ( reflection length ) ℓ T ( w ) of an element w ∈ W is defined by the smallest k such that w = t 1 t 2 · · · t k , where all t i are reflections. The absolute order ( reflection order ) ≤ T is defined by if and only if ℓ T ( u ) + ℓ T ( u − 1 w ) = ℓ T ( w ) . u ≤ T w Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Definition ( Armstrong) The m -divisible non-crossing partitions for a reflection group W are defined by � NC ( m ) ( W ) = ( w 0 ; w 1 , . . . , w m ) : w 0 w 1 · · · w m = c and � ℓ T ( w 0 ) + ℓ T ( w 1 ) + · · · + ℓ T ( w m ) = ℓ T ( c ) , where c is a Coxeter element in W . Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Definition ( Armstrong) The m -divisible non-crossing partitions for a reflection group W are defined by � NC ( m ) ( W ) = ( w 0 ; w 1 , . . . , w m ) : w 0 w 1 · · · w m = c and � ℓ T ( w 0 ) + ℓ T ( w 1 ) + · · · + ℓ T ( w m ) = ℓ T ( c ) , where c is a Coxeter element in W . In particular, NC (1) ( W ) ∼ = NC ( W ) , the “ordinary” non-crossing partitions for W . Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Combinatorial realisation in type A (Armstrong) � NC ( m ) ( W ) = ( w 0 ; w 1 , . . . , w m ) : w 0 w 1 · · · w m = c and � ℓ T ( w 0 ) + ℓ T ( w 1 ) + · · · + ℓ T ( w m ) = ℓ T ( c ) , Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Combinatorial realisation in type A (Armstrong) � NC ( m ) ( W ) = ( w 0 ; w 1 , . . . , w m ) : w 0 w 1 · · · w m = c and � ℓ T ( w 0 ) + ℓ T ( w 1 ) + · · · + ℓ T ( w m ) = ℓ T ( c ) , Example for m = 3 , W = A 6 Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Combinatorial realisation in type A (Armstrong) � NC ( m ) ( W ) = ( w 0 ; w 1 , . . . , w m ) : w 0 w 1 · · · w m = c and � ℓ T ( w 0 ) + ℓ T ( w 1 ) + · · · + ℓ T ( w m ) = ℓ T ( c ) , Example for m = 3 , W = A 6 (= S 7 ): Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Combinatorial realisation in type A (Armstrong) � NC ( m ) ( W ) = ( w 0 ; w 1 , . . . , w m ) : w 0 w 1 · · · w m = c and � ℓ T ( w 0 ) + ℓ T ( w 1 ) + · · · + ℓ T ( w m ) = ℓ T ( c ) , Example for m = 3 , W = A 6 (= S 7 ): w 0 = (4 , 5 , 6), w 1 = (3 , 6), w 2 = (1 , 7), and w 3 = (1 , 2 , 6). Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Combinatorial realisation in type A (Armstrong) � NC ( m ) ( W ) = ( w 0 ; w 1 , . . . , w m ) : w 0 w 1 · · · w m = c and � ℓ T ( w 0 ) + ℓ T ( w 1 ) + · · · + ℓ T ( w m ) = ℓ T ( c ) , Example for m = 3 , W = A 6 (= S 7 ): w 0 = (4 , 5 , 6), w 1 = (3 , 6), w 2 = (1 , 7), and w 3 = (1 , 2 , 6). Now “blow-up” w 1 , w 2 , w 3 : Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Combinatorial realisation in type A (Armstrong) � NC ( m ) ( W ) = ( w 0 ; w 1 , . . . , w m ) : w 0 w 1 · · · w m = c and � ℓ T ( w 0 ) + ℓ T ( w 1 ) + · · · + ℓ T ( w m ) = ℓ T ( c ) , Example for m = 3 , W = A 6 (= S 7 ): w 0 = (4 , 5 , 6), w 1 = (3 , 6), w 2 = (1 , 7), and w 3 = (1 , 2 , 6). Now “blow-up” w 1 , w 2 , w 3 : (7 , 16) (2 , 20) (3 , 6 , 18) Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Combinatorial realisation in type A (Armstrong) � NC ( m ) ( W ) = ( w 0 ; w 1 , . . . , w m ) : w 0 w 1 · · · w m = c and � ℓ T ( w 0 ) + ℓ T ( w 1 ) + · · · + ℓ T ( w m ) = ℓ T ( c ) , Example for m = 3 , W = A 6 (= S 7 ): w 0 = (4 , 5 , 6), w 1 = (3 , 6), w 2 = (1 , 7), and w 3 = (1 , 2 , 6). Now “blow-up” w 1 , w 2 , w 3 : (7 , 16) − 1 (2 , 20) − 1 (3 , 6 , 18) − 1 Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Combinatorial realisation in type A (Armstrong) � NC ( m ) ( W ) = ( w 0 ; w 1 , . . . , w m ) : w 0 w 1 · · · w m = c and � ℓ T ( w 0 ) + ℓ T ( w 1 ) + · · · + ℓ T ( w m ) = ℓ T ( c ) , Example for m = 3 , W = A 6 (= S 7 ): w 0 = (4 , 5 , 6), w 1 = (3 , 6), w 2 = (1 , 7), and w 3 = (1 , 2 , 6). Now “blow-up” w 1 , w 2 , w 3 : (1 , 2 , . . . , 21) (7 , 16) − 1 (2 , 20) − 1 (3 , 6 , 18) − 1 Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups Combinatorial realisation in type A (Armstrong) � NC ( m ) ( W ) = ( w 0 ; w 1 , . . . , w m ) : w 0 w 1 · · · w m = c and � ℓ T ( w 0 ) + ℓ T ( w 1 ) + · · · + ℓ T ( w m ) = ℓ T ( c ) , Example for m = 3 , W = A 6 (= S 7 ): w 0 = (4 , 5 , 6), w 1 = (3 , 6), w 2 = (1 , 7), and w 3 = (1 , 2 , 6). Now “blow-up” w 1 , w 2 , w 3 : (1 , 2 , . . . , 21) (7 , 16) − 1 (2 , 20) − 1 (3 , 6 , 18) − 1 = (1 , 2 , 21) (3 , 19 , 20) (4 , 5 , 6) (7 , 17 , 18) (8 , 9 , . . . , 16) . Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups 1 21 2 20 3 19 4 18 5 17 6 16 7 15 8 14 9 13 10 12 11 A 3-divisible non-crossing partition of type A 6 Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups 15 1 14 2 3 13 4 12 5 11 6 10 7 9 8 8 7 9 6 10 11 5 12 4 13 3 14 2 1 15 A 3-divisible non-crossing partition of type B 5 Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

m -divisible non-crossing partitions associated with reflection groups 15 1 14 2 3 13 4 12 5 11 6 10 18 17 7 9 8 8 16 16 7 9 17 18 6 10 11 5 12 4 13 3 14 2 1 15 A 3-divisible non-crossing partition of type D 6 Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

positive m -divisible non-crossing partitions We want positive m -divisible non-crossing partitions! Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

positive m -divisible non-crossing partitions We want positive m -divisible non-crossing partitions! These were defined by Buan, Reiten and Thomas, as an aside in “m-noncrossing partitions and m-clusters.” There, they constructed a bijection between the facets of the m -cluster complex of Fomin and Reading and the m -divisible non-crossing partitions of Armstrong. Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

positive m -divisible non-crossing partitions We want positive m -divisible non-crossing partitions! These were defined by Buan, Reiten and Thomas, as an aside in “m-noncrossing partitions and m-clusters.” There, they constructed a bijection between the facets of the m -cluster complex of Fomin and Reading and the m -divisible non-crossing partitions of Armstrong. The positive m -clusters are those which do not contain any negative roots. They are enumerated by the positive Fuß–Catalan numbers n � mh + d i − 2 Cat ( m ) + ( W ) := . d i i =1 Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving

positive m -divisible non-crossing partitions So: Christian Krattenthaler and Stump Positive m -divisible non-crossing partitions and their cylic sieving