Speaker
金久保有輝 (筑波大学大学院数理物質科学研究科)
Date
October 8, 10:00-10:50
Title
The inequalities defining polyhedral realizations and monomial realizations of crystal bases (slide)
Abstract
Nakashima and Zelevinsky invented `polyhedral realization', which is a kind of description of crystal bases \(B(\infty)\) as lattice points in some polyhedral convex cone. To construct the polyhedral realization, we need an infinite sequence \(\iota\) of indices. It is a natural problem to find an explicit form of inequalities defining the polyhedral convex cone. In this talk, we will give an explicit form of inequalities in terms of rectangular tableaux in the case the associated Lie algebra is classical type and \(\iota\) satisfies a condition. We also give a conjecture that the inequalities defining the polyhedral realization are obtained from monomial realizations which express the crystal bases of irreducible integrable highest weight representations as the set of Laurent monomials.