Author: Nicolás Libedinsky

I prove that for two positive weights one can produce a beautiful polytope (see the image at the side) and from the raw shape of that polytope one can produce the Lusztig's q-weight multiplicity of the two weights (also known as generalized Kostka-Foulkes polynomial, also known as Kazhdan-Lusztig polynomial of the associated elements in the affine Weyl group).

Joint with Jordan Ellenberg, David Plaza, José Simental, and Geordie Williamson, to appear in International Mathematics Research Notices. Inspired by permutations suggested by AlphaEvolve (an evolutionary coding agent developed by Google DeepMind), we discover massive hypercubes inside the Bruhat order of the symmetric group (the set of dyadically well-distributed permutations). This phenomenon has consequences for cluster algebras, Kazhdan–Lusztig theory, and moduli spaces of embeddings of Bruhat graphs, and provides a striking example of how AI can contribute to genuine mathematical discovery. From a mathematical point of view, the result is beautiful: the Bruhat interval defining this massive hypercube is described by a purely number-theoretic condition on permutations, a behavior completely unlike anything we had previously seen.

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, Revista Matemática Iberoamericana 42 (2026), no. 1, 187–228. This is the fifth paper on the development of a diagrammatic singular 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, Mathematical Research Letters 32 (2025), no. 6, 1845–1876. We produce an algorithmic construction of a reduced expression for any double coset.

Joint with Federico Castillo, Damian de la Fuente and David Plaza, Advances in Mathematics 499 (2026), Article 111048. DOI: 10.1016/j.aim.2026.111048. 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 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.