  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.
