## On the affine Hecke category for SL(3)

Joint with Leo Patimo. We find a surprisingly beautiful basis of the Hom spaces between indecomposable Soergel bimodules for SL(3) (something that we call "indecomposable light leaves").

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# For Mathematicians

## Highlighted Papers

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On the affine Hecke category for SL(3)

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My Education Manifesto

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Gentle introduction to Soergel bimodules I: The basics

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Pre-canonical bases on affine Hecke algebras

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Kazhdan-Lusztig polynomials and subexpressions

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p-Jones Wenzl idempotent

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Blob algebra approach to modular representation theory

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A non-perverse Soergel bimodule in type A

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The Anti-Spherical Category

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Our article with Jorge Soto-Andrade

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Research interests

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Invited talk in University of Sydney algebra seminar, 11th August 2017

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Plenary talk in “Representation Theory Day in Québec City”, 29th July 2017.

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Invited talk in “Math Congress of the Americas” in the session “Representations of Lie algebras”, 28th July 2017.

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Coorganizer (with Aaron Lauda and Joshua Sussan) of the session “Symmetry in algebra, topology, and physics” in the Math Congress of the Americas

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Invited stay in Sydney University, invited by Geordie Williamson, 5th July-25th August

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Indecomposable Soergel bimodules for Universal Coxeter groups

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Invited talk in the Conference “La resolución de Problemas, una competencia transversal”, 26th May 2017, Santiago, Chile.

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Sebastián Cea

Joint with Leo Patimo. We find a surprisingly beautiful basis of the Hom spaces between indecomposable Soergel bimodules for SL(3) (something that we call "indecomposable light leaves").

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July 1, 2020

I have been thinking about math education for more than fifteen years. An initiation rite for me was the following story. I was training three students for the math olympics in some school. They worked with me for one year without previous trainment. Then they went to the National Mathematical Olympiad in Chile and they obtained the three gold medals that were given that year. In that moment I was very moved and I realized that the method that I implemented (a method that has been evolving ever since) was interesting.

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September 20, 2017

** 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.

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September 1, 2017

Joint with Leo Patimo and David Plaza. 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.

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July 10, 2020

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.

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May 2, 2020

Joint work with Gastón Burrull and Paolo Sentinelli. . **Advances in Mathematics ** 352 (2019) 246-264. In this paper we find the 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.

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February 16, 2019

Joint work with David Plaza. **Proc. Lond. Math. Soc. ** Vol. 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...

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July 24, 2018

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.

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January 1, 2016

joint with Geordie Williamson, 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.

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November 11, 2017

Here is the first (and only until now) research article I did about these ideas, with Jorge Soto-Andrade On the…

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November 26, 2017

Most of my work revolves around some beautiful and central objects in representation theory called Soergel bimodules. I have produced a basis of morphisms between such objects that are called Light Leaves. I am fascinated by them. On the one hand, it is because of them that one can actually compute in Soergel bimodule-land, and on the other, they have some very rich combinatorial structure that is slowly unearthing. My favorite subjects are diagrammatical category theory, categorical braid group actions, modular representations of finite or algebraic groups, Kazhdan-Lusztig theory, categorifications and knot theory. Very nice.

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September 20, 2017

Invited talk in University of Sydney algebra seminar, 11th August 2017 University of Sydney Algebra Seminar Recent and forthcoming seminars…

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August 11, 2017

Plenary talk in "Representation Theory Day in Québec City", 29th July 2017. Schedule (Vachon building, Room 3840) All talks will…

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July 29, 2017

Invited talk in "Math Congress of the Americas" in the session "Representations of Lie algebras", 28th July 2017.

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July 28, 2017

Coorganizer (with Aaron Lauda and Joshua Sussan) of the session "Symmetry in algebra, topology, and physics" in the Math Congress of the…

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July 20, 2017

Invited stay in Sydney University, invited by Geordie Williamson, 5th July-25th August.

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July 5, 2017

joint with Ben Elias ; with an appendix by Ben Webster, **Trans. Amer. Math. Soc**.** **369 (2017), 3883-3910.
We find the explicit numbers for which "Soergel's conjecture in positive characteristic for Universal Coxeter systems" fail, i.e. we find explicitly the projectors to the indecomposables. These are clearly the most simple Coxeter systems (after the dihedral groups) but even in this case the situation is quite subtle. It would be amazing (for modular representation theory) to find similar results as the ones in this paper, but for Weyl groups or affine Weyl groups.

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June 1, 2017

Invited talk in the Conference "La resolución de Problemas, una competencia transversal", 26th May 2017, Santiago, Chile. SEMINARIO: "LA RESOLUCIÓN…

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May 26, 2017

Master student September 2016 – Present

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September 1, 2016