This unique book provides the first introduction to crystal base theory from the combinatorial
point of view. Crystal base theory was developed by Kashiwara and Lusztig from the …

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 …

In these lecture notes for a summer mini-course, we provide an exposition on quantum
groups and Hecke algebras, including (quasi) R-matrix, canonical basis, and q-Schur …

We prove that the weighted KLRW algebras of finite type, and their cyclotomic quotients, are
cellular algebras. The cellular bases are explicitly described using crystal graphs. As a …

We show how affine PBW bases can be used to construct affine MV polytopes, and that the
resulting objects agree with the affine MV polytopes recently constructed using either …

In the paper, we establish the global basis theory for the bosonic extension $\widehat
{\mathcal {A}} $ associated with an arbitrary generalized Cartan matrix. When $\widehat …

We construct a crystal basis for the negative half of the quantum group U associated to the
standard super Cartan datum of gl (m| 1), which is compatible with known crystals on Kac …

We study asymptotics of Tesler matrices using Kostant pictures, as well as partial orders on
these. We show that the Lusztig data partial order on Kostant pictures refines …

Comparing two perfect bases by Anne Dranowski A thesis submitted in conformity with the
requirements for the degree of Doctor of Page 1 Comparing two perfect bases by Anne …

Professor Xihua Cao (1920-2005) was a leading scholar at East China Normal University
(ECNU) and a famous algebraist in China. His contribution to the Chinese academic circle is …