Excursions into Algebra and Combinatorics at $q=0$

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 $\text{JJ}$-trivial monoids. We then discuss two-tensors of crystal bases for $U_q(\widetilde{{\mathfrak{s}\mathfrak{l}}_2})$, 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.