Promotion and cyclic sieving on families of SSYT

We examine a few families of semistandard Young tableaux, for which we observe the cyclic sieving phenomenon under promotion. The first family we consider consists of stretched hook shapes, where we use the cocharge generating polynomial as CSP-polynomial. The second family we consider consists of skew shapes, consisting of rectangles. Again, the charge generating polynomial together with promotion exhibits the cyclic sieving phenomenon. This generalizes earlier result by B. Rhoades and later B. Fontaine and J. Kamnitzer. Finally, we consider certain skew ribbons, where promotion behaves in a predictable manner. This result is stated in form of a bicyclic sieving phenomenon. One of the tools we use is a novel method for computing charge of skew semistandard tableaux, in the case when every number in the tableau occur with the same frequency

PP-strict promotion and BB-bounded rowmotion, with applications to tableaux of many flavors

We define P-strict labelings for a finite poset P as a generalization of semistandard Young tableaux and show that promotion on these objects is in equivariant bijection with a toggle action on B-bounded Q-partitions of an associated poset Q. In many nice cases, this toggle action is conjugate to rowmotion. We apply this result to flagged tableaux, Gelfand-Tsetlin patterns, and symplectic tableaux, obtaining new cyclic sieving and homomesy conjectures. We also show P-strict promotion can be equivalently defined using Bender-Knuth and jeu de taquin perspectives.Comment: 39 pages, 14 figure

Cyclic Sieving of Increasing Tableaux and small Schr\"oder Paths

An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schr\"oder paths. We demonstrate relations between the jeu de taquin for increasing tableaux developed by H. Thomas and A. Yong and the combinatorics of tropical frieze patterns. We then use this jeu de taquin to present new instances of the cyclic sieving phenomenon of V. Reiner, D. Stanton, and D. White, generalizing results of D. White and of J. Stembridge.Comment: 20 page

Noncommutative Schur P-functions and the Shifted Plactic Monoid.

This thesis is comprised of two related projects involving shifted tableaux. In the first project, we introduce a shifted analog of the plactic monoid of Lascoux and Schutzenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman's mixed insertion. Applications of the theory of the shifted plactic monoid include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux-Schutzenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more. In the second project, joint with T. K. Petersen, we show that the set of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon. The proofs rely heavily on the theory of the shifted plactic monoid.Ph.D.MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/77864/1/lserrano_1.pd

Cyclic sieving, promotion, and representation theory

We prove a collection of conjectures of D. White \cite{WComm}, as well as some related conjectures of Abuzzahab-Korson-Li-Meyer \cite{AKLM} and of Reiner and White \cite{ReinerComm}, \cite{WComm}, regarding the cyclic sieving phenomenon of Reiner, Stanton, and White \cite{RSWCSP} as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of $\mathbb{C}[x_{11}, ..., x_{nn}]$ due to Skandera \cite{SkanNNDCB}. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups and noncrossing partitions