70 research outputs found
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
Cyclic Sieving, Promotion, and Representation Theory
We prove a collection of conjectures due to Abuzzahab-Korson-Li-Meyer, Reiner, and White regarding the cyclic sieving phenomenon 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. 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, handshake patterns, and noncrossing partitions
Cyclic sieving, rotation, and geometric representation theory
We study rotation of invariant vectors in tensor products of minuscule
representations. We define a combinatorial notion of rotation of minuscule
Littelmann paths. Using affine Grassmannians, we show that this rotation action
is realized geometrically as rotation of components of the Satake fibre. As a
consequence, we have a basis for invariant spaces which is permuted by rotation
(up to global sign). Finally, we diagonalize the rotation operator by showing
that its eigenspaces are given by intersection homology of quiver varieties. As
a consequence, we generalize Rhoades' work on the cyclic sieving phenomenon.Comment: 16 page
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
Eulerian quasisymmetric functions and cyclic sieving
It is shown that a refined version of a q-analogue of the Eulerian numbers
together with the action, by conjugation, of the subgroup of the symmetric
group generated by the -cycle on the set of permutations
of fixed cycle type and fixed number of excedances provides an instance of the
cyclic sieving phenonmenon of Reiner, Stanton and White. The main tool is a
class of symmetric functions recently introduced in work of two of the authors.Comment: 30 page
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