Excursions into Algebra and Combinatorics at $q=0$

Details Excursions into Algebra and Combinatorics at $q=0$ Excursions into Algebra and Combinatorics at $q=0$

2011-08-22

arxiv.org/abs/1108.4379v1

Mathematics

Representation Theory

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

Scientific paper

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 $U_q(\tilde{\mathfrak{sl}_2})$, establishing a complementary result to one of Bandlow, Schilling, and Thi\'ery on affine crystals arising from promotion operators. Finally, we give a computer implementation of Stembridge's local axioms for simply-laced crystal bases.

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