UC Riverside

We study the structure of a Kashiwara crystal of simply-laced Cartan type \(\cd\) under an automorphism \(\sigma\), via a process known as folding.

We define the category of \(\sigma\)-foldable crystals, which is a monoidal category and admits a monoidal functor into the category of \(\cd^{\sigma\vee}\)-crystals.

Various properties of a foldable crystal---normality, Weyl group action, \emph{etc.}---can be transferred to its quotient modulo the \(\sigma\)-action.

On the other hand, the quotient of a foldable highest-weight crystal contains a new type of crystal, which we call multi-highest-weight.

We consider multi-highest-weight crystals \(B\) obtained as foldings of Kashiwara Littelmann crystals \(B(\lambda)\).

The structure of the highest-weight set of \(B\) is explained by certain subsets of the Weyl group we call the balanced parabolic quotients; in many cases the latter parameterizes a generating set for the highest-weight elements.

A balanced parabolic quotient relates the branching rules, Demazure crystals, and \(\sigma\)-action on \(B(\lambda)\), and is enumerated by a forest graph with self-similar components.