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Description
The Bruhat order on a Weyl group arises naturally from geometry, through inclusion relations among Schubert varieties, but it can also be defined combinatorially in the broader context of Coxeter systems. It is of major importance in Kazhdan-Lusztig theory, serving to define the remarkable Kazhdan-Lusztig basis and polynomials. Many fundamental questions about this partial order remain open; for instance, the combinatorial invariance conjecture asserts that Kazhdan-Lusztig polynomials are entirely determined by the poset structure of the underlying Bruhat intervals.
In this talk we will center on the study of lower Bruhat intervals (starting from the identity) for the case of affine Weyl groups and, specifically, on calculating their size. First, I will describe certain lower intervals by decomposing them in terms of lattice points in permutohedra. In turn, we will deduce a polynomial counting formula for the size of such intervals, expressed as a linear combination of the volumes of the faces of this polytope. Secondly, we will aim to generalize these results to almost all lower intervals: those arising from the lowest two-sided Kazhdan-Lusztig cell. The main object of study will be what we call "Paper Boat" sets, which will serve as fundamental building blocks of these lower intervals. This work is part of an on-going collaboration with Federico Castillo, Nicolás Libedinsky and David Plaza.