54 research outputs found
Promotion and evacuation on standard Young tableaux of rectangle and staircase shape
(Dual-)promotion and (dual-)evacuation are bijections on SYT(\lambda) for any
partition \lambda. Let c^r denote the rectangular partition (c,...,c) of height
r, and let sc_k (k > 2) denote the staircase partition (k,k-1,...,1). B.
Rhoades showed representation-theoretically that promotion on SYT(c^r) exhibits
the cyclic sieving phenomenon (CSP). In this paper, we demonstrate a promotion-
and evacuation-preserving embedding of SYT(sc_k) into SYT(k^{k+1}). This arose
from an attempt to demonstrate the CSP of promotion action on SYT(sc_k).Comment: 14 pages, typos correcte
Promotion on oscillating and alternating tableaux and rotation of matchings and permutations
Using Henriques' and Kamnitzer's cactus groups, Sch\"utzenberger's promotion
and evacuation operators on standard Young tableaux can be generalised in a
very natural way to operators acting on highest weight words in tensor products
of crystals.
For the crystals corresponding to the vector representations of the
symplectic groups, we show that Sundaram's map to perfect matchings intertwines
promotion and rotation of the associated chord diagrams, and evacuation and
reversal. We also exhibit a map with similar features for the crystals
corresponding to the adjoint representations of the general linear groups.
We prove these results by applying van Leeuwen's generalisation of Fomin's
local rules for jeu de taquin, connected to the action of the cactus groups by
Lenart, and variants of Fomin's growth diagrams for the Robinson-Schensted
correspondence
Monodromy and K-theory of Schubert curves via generalized jeu de taquin
We establish a combinatorial connection between the real geometry and the
-theory of complex Schubert curves , which are
one-dimensional Schubert problems defined with respect to flags osculating the
rational normal curve. In a previous paper, the second author showed that the
real geometry of these curves is described by the orbits of a map on
skew tableaux, defined as the commutator of jeu de taquin rectification and
promotion. In particular, the real locus of the Schubert curve is naturally a
covering space of , with as the monodromy operator.
We provide a local algorithm for computing without rectifying the
skew tableau, and show that certain steps in our algorithm are in bijective
correspondence with Pechenik and Yong's genomic tableaux, which enumerate the
-theoretic Littlewood-Richardson coefficient associated to the Schubert
curve. We then give purely combinatorial proofs of several numerical results
involving the -theory and real geometry of .Comment: 33 pages, 12 figures including 2 color figures; to appear in the
Journal of Algebraic Combinatoric
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
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
Some remarks on sign-balanced and maj-balanced posets
Let P be a poset with elements 1,2,...,n. We say that P is sign-balanced if
exactly half the linear extensions of P (regarded as permutations of 1,2,...,n)
are even permutations, i.e., have an even number of inversions. This concept
first arose in the work of Frank Ruskey, who was interested in the efficient
generation of all linear extensions of P. We survey a number of techniques for
showing that posets are sign-balanced, and more generally, computing their
"imbalance." There are close connections with domino tilings and, for certain
posets, a "domino generalization" of Schur functions due to Carre and Leclerc.
We also say that P is maj-balanced if exactly half the linear extensions of P
have even major index. We discuss some similarities and some differences
between sign-balanced and maj-balanced posets.Comment: 30 pages. Some inaccuracies in Section 3 have been corrected, and
Conjecture 3.6 has been adde
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