Joint with Gastón Burrull and Rodrigo Villegas. In this paper, we show that classifying Bruhat intervals (up to poset isomorphism) is not the nightmare it first appears to be. Our proof may not win a beauty contest :) but the result itself turns out to be surprisingly simple and elegant.
Joint with Ben Elias, Hankyung Ko and Leonardo Patimo. This is the fifth paper on the development of a singular diagrammatic Hecke theory. Here we introduce a generalization of the twisted Leibniz rule for the Demazure operator associated to any atomic double coset. This is equivalent to a polynomial forcing property for singular Soergel bimodules.
Joint work with Federico Castillo, Damian de la Fuente, and David Plaza. We introduce the notion of a Paper Boat: a new combinatorial object that tessellates the lower intervals of elements in the lowest two-sided Kazhdan–Lusztig cell. Using Paper Boats and convex geometry, we obtain formulas for the cardinalities of these intervals and show that they are given by (quasi-)polynomials. We further conjecture that, in fact, they are always genuine polynomials. As a byproduct of this work, we also discovered a natural partition of the group that is particularly well adapted to studying polynomiality, and which appears to have interesting applications in its own right.
Joint with Ben Elias, Hankyung Ko and Leonardo Patimo, to appear in Mathematical Research Letters. We produce an algorithmic construction of a reduced expression for any double coset.
Joint with Ben Elias, Hankyung Ko and Leonardo Patimo, to appear in the Journal of the European Mathematical Society . For any Coxeter system, we introduce the concept of singular light leaves (finally!!). Really fun stuff.
Joint with Federico Castillo, Damian de la Fuente and David Plaza. For affine Weyl groups and elements associated to dominant coweights, we give a convex geometry formula for the size of the corresponding lower Bruhat intervals (this says in particular that it is a polynomial in several variables). Extensive computer calculations for affine Weyl groups have led us to believe that a similar formula exists for all lower Bruhat intervals. We also believe that the cardinality of all Bruhat intervals is given by some family of quasi-polynomials.
Joint with Ben Elias, Hankyung Ko and Leonardo Patimo. This is the second paper on the development of a singular Coxeter theory. Here we give a singular analogue of the usual subexpression version of the Bruhat order.
Joint with Ben Elias, Hankyung Ko and Leonardo Patimo. This is the first of a series of papers intended to advance in the development of a singular (i.e. double cosets) theory of Coxeter groups, Hecke algebras, actions of groups on polynomial rings and the Hecke category. Two milestones of this long-term project would be to produce singular light leaves and singular Soergel calculus. Here, we introduce a Demazure operator for any double coset. We prove several results about them, but the crucial thing for us is that they give a criterion for ensuring the proper behavior of singular Soergel bimodules.
Joint with Leonardo Patimo, Selecta Mathematica 29 (2023), no. 4, Paper No. 64. We find a surprisingly beautiful basis of the Hom spaces between indecomposable Soergel bimodules for SL(3) (something that we call indecomposable light leaves.
Accepted for publication in Journal of the Indian Institute of Science 102, No. 3, 907-946 (2022). It is an introduction/survey of representation theory with lots of humor, open questions, and extremely smart insights :)
Joint with Gastón Burrull and David Plaza, International Mathematics Research Notices 10 (2023) 8903–8933. We prove the combinatorial invariance conjecture (by G. Lusztig and M. Dyer in the eighties) for the affine A2. This is the first infinite group with non-trivial Kazhdan-Lusztig polynomials where this fascinating conjecture is proved.
Joint with Leonardo Patimo and David Plaza, Advances in Mathematics 399 (2022). For any affine Weyl group, we introduce the pre-canonical bases, a set of bases of the spherical Hecke algebra that interpolates between the standard basis and the canonical basis. Thus we divide the hard problem of calculating Kazhdan-Lusztig polynomials (or q-analogues of weight multiplicities) into a finite number of much easier problems.
joint with Geordie Williamson, Advances in Mathematics 405 (2022). We prove that (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients for ANY Coxeter system and ANY choice of a parabolic subgroup, thus generalizing to the parabolic setting the central result of The Hodge theory of Soergel bimodules by Elias and Williamson. We also prove a monotonicity conjecture of Brenti. The new techniques were used by Williamson and Lusztig to calculate many new elements of the p-canonical basis and thus make the Billiards conjecture. Along the way, we introduce the anti-spherical light leaves.
Joint with Geordie Williamson, Journal of Algebra 568 (2021) 181-192. When Soergel's conjecture is satisfied, we produce (finally!) the canonical light leaves, that do not depend on choices. This gives a new approach towards finding a combinatorial interpretation of Kazhdan-Lusztig polynomials.
Joint work with David Plaza. Proceedings of the London Mathematical Society 121 (2020) Issue3, 656-701. We conjecture (and prove the "graded degree part") an equivalence between the type A affine Hecke category in positive characteristic and a certain blob category that we introduce as a quotient of KLR algebras. This conjecture has been proved recently in an amazing paper by Chris Bowman, Anton Cox, Amit Hazi!! It opens lots of questions...
Joint work with Gastón Burrull and Paolo Sentinelli. Advances in Mathematics 352 (2019) 246-264. In this paper we introduce the p-Jones Wenzl idempotent, a characteristic p analogue of the classical Jones-Wenzl idempotent. We hope this to be a building block for the p-canonical basis as sl_2 is a building block for the representation theory of semi-simple Lie algebras.
Sao Paulo Journal of Mathematical Sciences, 13(2) (2019), 499-538. This paper is the first of a series of introductory papers on the fascinating world of Soergel bimodules. It is combinatorial in nature and should be accessible to a broad audience. We introduce the Forking path conjecture.
joint with Geordie Williamson, Comptes Rendus Mathematique Vol 355 (2017) Issue 8, 853-858. We prove that there are indecomposable Soergel bimodules (in type A) having negative degree endomorphisms. This is quite surprising and proves the existence of a non-perverse parity sheaf in type A.
joint with Ben Elias ; with an appendix by Ben Webster, Transactions of the American Mathematical Society 369 (2017), 3883-3910. We introduce the multicolored Temperley-Lieb 2-category. Using it, we find the explicit numbers for which "Soergel's conjecture in positive characteristic for Universal Coxeter systems" fail.
Advances in Mathematics (2015) 772-807. I introduce the double leaves basis and with it I prove that Lusztig's conjecture reduces to a problem about the light leaves. Using the result in Section 4.3 of this paper Geordie Williamson disproved Lusztig's conjecture! The counterexamples grow exponentially in the Coxeter number. Here is Geordie's paper
