Composite Sieving Techniques: Dihedral Action on Cluster Complexes Zachary Stier Julian Wellman Zixuan Xu UMN REU July 25, 2019 Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 1 / 29
Outline ∗ Motivation : cyclic and dihedral sieving ∗ Our results : dihedral sieving on cluster complexes ∗ Future directions Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 2 / 29
q - and q, t -analogues Definition n − 1 n − 1 � � q i t n − 1 − i q i { n } q,t := [ n ] q := { n } q, 1 = i =0 i =0 n n � � { n } ! q,t := { n } q,t [ n ]! q := { n } ! q, 1 = [ n ] q i =1 i =1 � n � { n } ! q,t � n � � n � [ n ]! q := := = k { k } ! q,t { n − k } ! q,t k k [ k ]! q [ n − k ]! q q, 1 q,t q Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 3 / 29
A strange behavior � 14 � � 14 � 1 1 = 6 and = 0 [8] ω 3 7 [8] ω 4 7 ω 3 ω 4 9 9 9 9 What’s going on? Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 4 / 29
Cyclic sieving Definition (Reiner–Stanton–White ’04) If X is a finite set acted on by a cyclic group C n = � r � , and X ( q ) is a polynomial in q , then the pair ( X C n , X ( q )) has the cyclic sieving � phenomenon (CSP) if for all ℓ ∈ [ n ], # { x ∈ X : r ℓ x = x } = X ( ω ℓ n ) . Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 5 / 29
Examples of cyclic sieving � � � n � ∗ Let X be the k -subsets of [ n ]. Then X C n , exhibits CSP. � k q [Reiner–Stanton–White ’04] � � � n − k +1 � ∗ Let X be the k -multisubsets of [ n ]. Then X C n , � k q exhibits CSP. [Reiner–Stanton–White ’04] ∗ Let X be the set of k -angulations of an n -gon. Then � � � ( k − 1) m 1 exhibits CSP, for m := n − 2 � X C n , k − 2 . [Eu–Fu ’06] � [ m ] q m q Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 6 / 29
k -angulations of an n -gon It is easily verified that such a dissection exists iff n ≡ 2 mod ( k − 2). Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 7 / 29
Another strange behavior? Cat n ( ω 3 9 , ω − 3 Cat n ( ω 4 9 , ω − 4 9 ) = 6 and 9 ) = 0 and Cat n (1 , − 1) = 5 What’s this Cat n ? Is there something else going on? Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 8 / 29
Dihedral sieving Definition (Rao–Suk ’17) If X is a finite set acted on by a dihedral group I 2 ( n ) = � r 1 , r 2 � for odd n , and X ( q, t ) is a symmetric polynomial in q and t , then the pair ( X I 2 ( n ) , X ( q, t )) has the dihedral sieving phenomenon (DSP) if � � { ω k , ω k } g a rotation for all g ∈ I 2 ( n ) with { λ 1 , λ 2 } = g a reflection, { 1 , − 1 } # { x ∈ X : gx = x } = X ( λ 1 , λ 2 ) . Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 9 / 29
Examples of dihedral sieving � � � n � ∗ Let X be the k -subsets of [ n ]. Then X I 2 ( n ) , ( q, t ) � k exhibits DSP for odd n . [Rao–Suk ’17] ∗ Let X be the k -multisubsets of [ n ]. Then � � � n − k +1 � X I 2 ( n ) , ( q, t ) exhibits DSP for odd n . [Rao–Suk ’17] � k ∗ Let X be the set of triangulations of an n -gon. Then � � I 2 ( n ) , Cat n ( q, t ) exhibits DSP for odd n . [Rao–Suk ’17] X � Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 10 / 29
Just triangulations? Question: Does anything stop us from obtaining DSP for k -angulations? Answer: No. Theorem (REU ’19) Let X be the set of k -angulations of an n -gon. Then � � I 2 ( n ) , Cat k X n ( q, t ) exhibits DSP for all odd n and k . � Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 11 / 29
Cat k n ( q, t ) ( ms + 1 , m ) (0 , 0) � Cat k q area ( λ ) t area ( sweep ( λ )) n ( q, t ) := λ � ( k − 1) m � 1 Cat k n ( ω, ω ) = [ m ] ω m − 1 ω Cat k n ( − 1 , 1) ≡ # even area paths − # odd area paths Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 12 / 29
Cat k n ( q, t ) ( ms + 1 , m ) (0 , 0) � Cat k q area ( λ ) t area ( sweep ( λ )) n ( q, t ) := λ � ( k − 1) m � 1 Cat k n ( ω, ω ) = [ m ] ω m − 1 ω Cat k n ( − 1 , 1) ≡ # even area paths − # odd area paths Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 12 / 29
Raney numbers r � pm + r � R p,r ( m ) := pm + r m m − 1 � R p, 1 ( m ) = R p, 1 ( i ) R p,p − 1 ( m − 1 − i ) [Zhou–Yan ’17] i =0 m � R p,r ( m ) = R p,r ( i ) R p,r − 1 ( m − i ) [Zhou–Yan ’17] i =0 Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 13 / 29
DSP for k -angulations Theorem For odd k > 3 and m , � m − 1 � m − 1 Cat k 2 R s +1 , s +1 n (1 , − 1) = ( − 1) . 2 2 Proof sketch. We use recursion on Young diagrams of the shape shown before. We ( − 1) area ( λ ) . define D s ( ℓ, m ) = � λ Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 14 / 29
DSP for k -angulations Proof sketch, cont. m − 2 � ( − 1) y +1 D s (1 , y ) D s (2 y − 1 , m − y − 2) D s (1 , m ) = y =0 follows by considering recursion on the SW-most marker contained above given a path. y m − 1 s s s · · · s − 1 s . . . . ... ... . . . . . . . . s y 3 s − 1 s y 2 s − 1 s y 1 s Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 14 / 29
DSP for k -angulations Proof sketch, cont. m � ( − 1) ( m +1) y D s ( ℓ − 2 , y ) D s (1 , m − y ) D s ( ℓ, m ) = y =0 follows by considering recursion on the SW-most marker contained above given a path. (Note the different marker configuration.) y m − 1 ℓ s s · · · s − 2 . . . ... . . . . . . s s − 2 y 2 ℓ s − 2 y 1 ℓ y 0 ℓ − 2 Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 14 / 29
DSP for k -angulations Proof sketch, cont. D s ( ℓ, m ) = 0 for odd ℓ and m , so we can rewrite the recurrences for � m � D s ( ℓ, m ) to match those for R s +1 , ℓ +1 � 2 2 The final case of is k = 3—triangulations, proved in in Rao–Suk ’17. Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 14 / 29
Generalizing k -angulations? Question: Is there another layer of generality to look at? Answer: Yes. k -angulations arise as maximal clusters in ∆(Φ( A n )). Let’s go into what that means. Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 15 / 29
Root systems Take a Coxeter group W with root system Φ = Φ( W ), simple roots Π, and positive roots Φ + . Let Φ − 1 := Φ + ⊔ − Π. Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 16 / 29
Positive root posets and Cat W ( q, t ) � q, 1 � [ h + d i ] q = q � � Cat W ( q, 1) = q � � q | J | Cat W [ d i ] q q i J ∈ I ( P ) Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 17 / 29
Example: W = A 2 ( − 1) 0 ( − 1) 1 ( − 1) 1 ( − 1) 2 ( − 1) 3 ( − 1) | J | = 1 − 1 − 1 + 1 − 1 = − 1 = ± 1 � � J ∈ I ( P ) Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 18 / 29
Cluster complexes Clusters are maximal sets of mutually- compatible roots, forming the cluster complex ∆(Φ). Further, there is the action � τ − , τ + � ∼ Φ − 1 = I 2 ( n ) . � ∆(Φ) generalizes to ∆ ( s ) (Φ), with a corresponding poset and Cat ( s ) W . Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 19 / 29
Main results In this language, we can reformulate the previous theorem: Theorem (REU ’19) I 2 ( n + 2) , Cat ( s ) The pair (∆ ( s ) (Φ( A n − 1 )) A n − 1 ( q, t )) exhibits dihedral � sieving for all odd n and s . We also prove: Theorem (REU ’19) The pair (∆(Φ) I 2 ( h + 2) , Cat W ( q, t )) exhibits dihedral sieving for � any root system Φ = Φ( W ) when h = max { d i } is odd. Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 20 / 29
Type A ∆ ( s ) (Φ( A n − 1 )) ← → { k -angulations of ( n + 2)-gon } Since the action of � τ − , τ + � is dihderal, for odd n —done! Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 21 / 29
Type B cluster complexes Clusters of ∆(Φ( B n − 1 )) correspond to centrally symmetric k -angulations of a 2 n -gon with a diameter. It is evident that no such k -angulation is fixed under a reflection. Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 22 / 29
Type B cluster complexes Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 23 / 29
Type B cluster complexes The positive root poset of B n is the trapezoid poset T n, 2 n . Let the triangle poset be T n . Lemma ( − 1) | J | = 0 ( − 1) | J | = 0 . � � and hence J ∈ I ( T n ) J ∈ I ( T n, 2 n ) Proof sketch. Induction on n and the number of minimal elements included in a given order ideal. � Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 24 / 29
Type D cluster complexes Clusters of ∆(Φ( D n − 1 )) correspond to centrally symmetric k -angulations of a 2 n -gon with colored diameters that may intersect the same color. Reflection switches their color. It is evident that no such k -angulation is fixed under a reflection. We can show that the desired polynomial vanishes in the same way as in Type B . Stier, Wellman, Xu Dihedral Sieving Phenomena July 25, 2019 25 / 29
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