Excursions into Algebra and Combinatorics at $q=0$

Abstract

We explore combinatorics associated with the degenerate Hecke algebra at $q=0$, obtaining a formula for a system of orthogonal idempotents, and also exploring various pattern avoidance results. Generalizing constructions for the 0-Hecke algebra, we explore the representation theory of $\JJ$-trivial monoids. We then discuss two-tensors of crystal bases for $U_q(\widetilde{{\mathfrak{sl}}_2})$, establishing a complementary result to one of Bandlow, Schilling, and Thiéry on affine crystals arising from promotion operators. Finally, we give a computer implementation of Stembridge's local axioms for simply-laced crystal bases.

92 pages, 13 figures. PHd Dissertation accepted at the University of California on July 15th, 2011. arXiv admin note: text overlap with arXiv:1012.1361

References in corpus (5)

• Kirillov--Reshetikhin crystals for nonexceptional types
• K-theory Schubert calculus of the affine Grassmannian
• Primitive orthogonal idempotents for R-trivial monoids
• On Kiselman quotients of 0-Hecke monoids
• Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras