Hypercube decompositions and the Combinatorial Invariance Conjecture

Given two elements u and v of the symmetric group (or a more general Coxeter group), Kazhdan and Lusztig associate a polynomial P_uv which describes the intersection cohomology of the Schubert variety X_v and, through the (proven) Kazhdan-Lusztig Conjecture, gives character formulas in the representation theory of gl_n. It was observed by Dyer and Lusztig that whenever the intervals [u,v] and [u’,v’] in Bruhat order are isomorphic as posets, seemingly there is also an equality P_uv = P_u’v’. This has come to be known as the Combinatorial Invariance Conjecture. Recently, Blundell-Buesing-Davies-Veličković-Williamson posed a conjecture which, if true for all intervals, would imply the Combinatorial Invariance Conjecture for the symmetric group. Their conjecture gives an explicit recurrence for P_{uv} using hypercube decompositions. We prove the BBDVW conjecture for lower intervals (intervals of the form [id, v]). We also study the family of Kazhdan-Lusztig R-polynomials, which count points in a Richardson variety over a finite field and can be used to compute P_uv. Using hypercube decompositions, we establish Combinatorial Invariance for the R-polynomials of elementary intervals, which include lower intervals and intervals isomorphic to them.

This is joint work with Christian Gaetz.