Invariant tensors and the cyclic sieving phenomenon

TL;DR

This paper constructs numerous examples of the cyclic sieving phenomenon using the representation theory of semi-simple Lie algebras, linking crystal bases, promotion operators, and symmetric group actions.

Contribution

It introduces a general framework for realizing the cyclic sieving phenomenon via invariant tensors and crystal bases in Lie algebra representations.

Findings

Constructed cyclic sieving triples from crystal bases and promotion operators.

Connected cyclic sieving to invariant tensors in semi-simple Lie algebra representations.

Applied framework to rectangular tableaux using the defining representation of SL(n).

Abstract

We construct a large class of examples of the cyclic sieving phenomenon by expoiting 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.