Invariant tensors and the cyclic sieving phenomenon

Bruce W. Westbury¹

¹University of Warwick

The electronic journal of combinatorics, Tome 23 (2016) no. 4

Résumé

We construct a large class of examples of the cyclic sieving phenomenon by exploiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the associated Kashiwara crystal. For $r\ge 0$, the triple $(X,c,P)$ which exhibits the cyclic sieving phenomenon is constructed as follows: the set $X$ is the set of isolated vertices in the crystal $\otimes^rB$; the map $c\colon X\rightarrow X$ is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial $P$ is the fake degree of the Frobenius character of a representation of $\mathfrak{S}_r$ related to the natural action of $\mathfrak{S}_r$ on the subspace of invariant tensors in $\otimes^rM$. Taking $M$ to be the defining representation of $\mathrm{SL}(n)$ gives the cyclic sieving phenomenon for rectangular tableaux.

Détail

DOI : 10.37236/4569

Classification : 05E10, 17B20, 17B10
Mots-clés : cyclic sieving phenomenon, promotion, crystal graphs