Dihedral Sieving Phenomena Sujit Rao, Joe Suk 31 July 2017 1/14 - PowerPoint PPT Presentation

Dihedral Sieving Phenomena Sujit Rao, Joe Suk 31 July 2017 1/14

Outline Cyclic sieving Sieving for general groups Sieving for dihedral groups Examples of dihedral sieving Future work 2/14

Cyclic Sieving Phenomenon (CSP) Definition X be a finite set, X p q q P N r q s , and ω : C n Ñ C ˆ be an Let C n ý embedding. Then p X , X p q q , C q exhibits cyclic sieving if @ c P C : | X p q q| q “ ω p c q “ |t x P X : c p x q “ x u| . Example “ n ` k ´ 1 § pt k -multisubsets of r n su , ‰ q , C n q k “ n § pt k -subsets of r n su , ‰ q , C n q k § pt noncrossing partitions of n -gon u , C n p q q , C n q § pt triangulations of a regular n -gon u , C n ´ 2 p q q , C n q ˆ" dissections of n -gon * ˙ “ n ` k “ n ´ 3 1 ‰ ‰ , q , C n § r n ` k s q k ` 1 k with k diagonals q ˆ" noncrossing partitions of * ˙ , N p n , k ; q q , C n . § n -gon with n ´ k parts 3/14

k -multisubsets Proposition (Reiner-Stanton-White 2004) Let V be a f.d. GL n p C q -rep. Assume C permutes a basis t v x u x P X . If X p q q “ χ ρ p 1 , q , . . . , q n ´ 1 q “ Tr diag p 1 , . . . , q n ´ 1 q : V Ñ V ` ˘ . then p X , X p q q , C q has CSP. Corollary (Reiner-Stanton-White 2004) Let V “ Sym k p C N q . If λ “ p k q $ k , then „ N ` k ´ 1  χ V p 1 , q , . . . , q N ´ 1 q “ s λ p 1 , q , . . . , q N ´ 1 q “ k q “ N ` k ´ 1 ‰ and hence pt k -multisubsets of r N su , q , C n q has CSP. k 4/14

Equivalent Definition of Cyclic Sieving Proposition (Reiner-Stanton-White 2004) Consider p X , X p q q , C q . Let A X be a graded C -vector space à A X “ A X , i i ě 0 with dim C A X , i q i “ X p q q . ÿ i ě 0 A X , i by c ¨ v “ ω p c q i v . Then p X , X p q q , C q has CSP Define C ý iff A X – C X . 5/14

Representation Ring Definition The representation ring of G with coefficients in R is Rep p G ; R q “ R rt iso. classes of f.d. G -reps us{p I ` J q where I “ ptr U ‘ V s ´ pr U s ` r V squq J “ ptr U b V s ´ r U sr V suq . Fact An isomorphism Rep p G ; C q Ñ ClFun p G q is given by r V s ÞÑ χ V . Fact Rep p G ; R q has an R -basis consisting of irreducible representations. 6/14

Cyclic Sieving Rephrased Example An isomorphism Z r q s{p q n ´ 1 q Ñ Rep p C n ; Z q is given by q ÞÑ ω n . Proposition (Reiner-Stanton-White 2004) p X , X p q q , C n q has cyclic sieving ô C X “ X p ω n q in Rep p C n ; Z q . Definition Let ρ 1 , . . . , ρ k be representations of G . Let G X and ý X p q 1 , . . . , q k q P C r q 1 , . . . , q k s . Then p X , X p q 1 , . . . , q k q , p ρ 1 , . . . , ρ k q , G q has G -sieving if C X “ X p ρ 1 , . . . , ρ k q in Rep p G ; C q . 7/14

Examples Example (Cyclic Sieving) G “ C n and ρ 1 “ ω n for an embedding ω n : C n Ñ C ˆ . Definition (Barcelo-Reiner-Stanton 2007) Let G “ C n ˆ C m and ω n : C n Ñ C ˆ , ω m : C m Ñ C ˆ be embeddings. Let X p t , q q P Z r t , q s . Then p X , X p t , q q , C n ˆ C m q has bicyclic sieving if X p ω n p c q , ω m p c 1 qq “ |t x P X : p c , c 1 q x “ x u| . Example (Bicyclic Sieving) G “ C n ˆ C m , ρ 1 “ ω n b 1 m and ρ 2 “ 1 n b ω m . 8/14

Irreducible Representations of I 2 p n q Let the dihedral group of order 2 n be ˇ r n “ s 2 “ e , rs “ sr ´ 1 D @ ˇ I 2 p n q “ r , s . n odd ✶ , det, z 1 , . . . , z p n ´ 1 q{ 2 . n even ✶ , det, χ a , χ a ¨ det, z 1 , . . . , z p n ´ 2 q{ 2 . „ cos p 2 π k ´ sin p 2 π k  n q n q z i p r q “ sin p 2 π k cos p 2 π k n q n q „ 1  0 z i p s q “ 0 ´ 1 Fact z i “ Ind I 2 p n q ω i . C n 9/14

Properties of Fibonomial Coefficients Definition (Amdeberhan, Chen, Moll, Sagan) t 0 u s , t “ 0 t 1 u s , t “ 1 t n ` 2 u s , t “ s t n ` 1 u s , t ` t t n u s , t . Proposition (Amdeberhan, Chen, Moll, Sagan) ? ? Let X “ s ` s 2 ` 4 t and Y “ s ´ s 2 ` 4 t . Then 2 2 r n s q “ t n u s , t | s “ q ` 1 , t “´ q t n u s , t “ Y n ´ 1 r n s q | q “ X { Y ˇ " n * „ n  ˇ “ Y k p n ´ k q . ˇ k k ˇ s , t q ˇ q “ X { Y 10/14

k -multisubsets Proposition Let n be odd, V a f.d. GL n p C q -rep. Assume I 2 p n q permutes a basis t v x : x P X u . Let p P Z r x , y s be unique such that p p a ` b , ab q “ χ V p a n ´ 1 , a n ´ 2 b , . . . , ab n ´ 2 , b n ´ 1 q Then p X , p , p z 1 , ´ det q , I 2 p n qq exhibits dihedral sieving. Corollary ´ ´ ¯ ¯ r n s Let n be odd and X “ . Then k ˜ ¸ " n ` k ´ 1 * , p z 1 , ´ det q , I 2 p n q X , k s , t exhibits dihedral sieving. 11/14

Comparison of generating functions C n ρ 1 “ ω n I 2 p n q ρ 1 “ z 1 , ρ 2 “ ´ det I 2 p n q n odd C n “ n � n ‰ ( t k -subsets of r n su k k “ n ` k ´ 1 � n ` k ´ 1 ‰ ( t k -multisubsets of r n su k k “ 2 n � 2 n 1 1 ‰ ( t NC partitions of r n su r n ` 1 s n t n ` 1 u n " * ‰“ n (� n NC partitions of r n s “ n � n 1 q k p k ` 1 q 1 ‰ ( r n s k k ` 1 t n u k k ` 1 with n ´ k blocks 12/14

Possibly useful polynomials for even n t 0 u s , t , a “ 0 t 1 u s , t , a “ 1 t 2 u s , t , a “ s # sa t n ´ 1 u ` t t n ´ 2 u n is odd t n u s , t , a “ s t n ´ 1 u ` t t n ´ 2 u n is even . with substitution p s , t , a q “ p z 1 ` b { n , ´ det , 1 ´ b { 4 q gives $ “ n t r ℓ , r n ´ ℓ u ‰ k q “ ξ 2 ℓ ’ " n * ’ n & “ n t sr , sr 3 , sr 5 , . . . u ‰ “ k q “ ξ 2 k s , t , a “ n ´ 2 “ n ´ 2 ’ “ n t sr 2 , sr 4 , . . . u ‰ ‰ ‰ q “ ξ 2 ` 2 ξ 2 ` ’ % k k ´ 1 k ´ 2 q “ ξ 2 13/14

Further Dihedral Actions § Triangulations and dissections of an n -gon § Rhoades’s promotion-evacuation action on rectangular tableaux § I 2 p 4 q on alternating sign matrices 14/14

Acknowledgments This research was carried out as part of the 2017 summer REU program at the School of Mathematics, University of Minnesota, Twin Cities, and was supported by NSF RTG grant DMS-1148634. The authors would like to thank Victor Reiner, Pavlo Pylyavskyy, and Benjamin Strasser for their mentorship and support. 15/14