Permutations $w$ in $S_n$ for which the (type-A) Schubert variety $\Omega_w$ is smooth are characterized by avoidance of the patterns 3412 and 4231.  The smaller family of codominant permutations, those avoiding the pattern 312, seems to explain a lot about character evaluations at Kazhdan-Lusztig basis elements $C’_w(q)$ of the (type-A) Hecke algebra. In particular, for every Hecke algebra character $\chi$, and every 3412-, 4231-avoiding permutation $w$, there exists a codominant permutation $v$ such that $\chi(C’_w(q)) = \chi(C’_v(q)$. Moreover, these character evaluations can be computed by playing simple games with unit interval orders $P = P(v)$ corresponding to the codominant permutations.  We generalize these facts to the hyperoctahedral group $B_n$ using signed pattern avoidance and an appropriate analog of unit interval orders.

Speaker’s webpage: https://www.lehigh.edu/~mas906/