Ben Salisbury

COMBINATORIAL DESCRIPTIONS OF THE CRYSTAL STRUCTURE ON CERTAIN PBW BASES

Using the theory of PBW bases, one can realize the crystal B(∞) for any semisimple Lie algebra over C using Kostant partitions as the underlying set. In fact there are many such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. Here we show that, for certain reduced expressions, the crystal operators can also be described by a much simpler bracketing rule. We give conditions describing these reduced expressions, and show that there is at least one example in every type except possibly E8, F4 and G2. We then discuss some examples.

A rigged configuration model for B (

We describe a combinatorial realization of the crystals B ( ∞ ) and B ( λ ) using rigged configurations in all symmetrizable Kac-Moody types up to certain conditions. This includes all simply-laced types and all non-simply-laced finite and affine types.

Connecting Marginally Large Tableaux and Rigged Configurations via Crystals

We show that the bijection from rigged configurations to tensor products of Kirillov-Reshetikhin crystals extends to a crystal isomorphism between the B(∞)

Rigged configurations and the \(*\)-involution

B(\infty )

PBW bases and marginally large tableaux in types B and C

models given by rigged configurations and marginally large tableaux.

The weight function for monomial crystals of affine type

We give an explicit description of the

PBW Bases and Marginally Large Tableaux in Type D

*

Virtual crystals and Nakajima monomials

∗-involution on the rigged configuration model for