Submit progress

on Tableaux problem 31

The statement

31.[Bren2024a]*Conj. 1.4 Lusztig’s “Combinatorial Invariance Conjecture” about the Kazhdan–Lusztig polynomials $P_{x,y}(q)$ : Let $(W_{1},S_{1}),(W_{2},S_{2})$ be two Coxeter systems, and $u\leq v$ in $W_{1}$ , $w\leq z$ in $W_{2}$ (in the strong Bruhat order) be such that $[u,v]\simeq[w,z]$ (the intervals in the Strong Bruhat order are isomorphic as posets). Then prove

P_{u,v}(q)=P_{w,z}(q).

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.