Highlighted Papers

On the affine Hecke category

Joint with Leo Patimo. We give a complete (and surprisingly simple) description of the affine Hecke category for SL(3) in characteristic zero. More precisely, we calculate the Kazhdan-Lusztig polynomials, give a recursive formula for the projectors defining indecomposable objects and, for each coefficient of a Kazhdan-Lusztig polynomial, we produce a set of morphisms with such a cardinality.

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p-Jones Wenzl idempotent

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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Blob algebra approach to modular representation theory

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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The Anti-Spherical Category

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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Indecomposable Soergel bimodules for Universal Coxeter groups

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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Standard objects in 2-braid groups

Joint with Geordie WilliamsonProc. London. Math. Soc. 109 (2014), no. 5, 1264-1280. For any Coxeter system, we establish the existence of analogues of standard and costandard objects in 2-braid groups, thus proving the conjecture that Rouquier stated in the ICM 2006. This result was a key step for the proof by Elias and Williamson of Kazhdan-Lusztig conjectures

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New bases of some Hecke algebras via Soergel bimodules 

Advances in Math. 228 (2011) 1043-1067. This is a first attempt to find explicitly Soergel indecomposable bimodules for extra-large Coxeter systems. This is very linked with my "Forking path conjecture" (see the paper "Gentle Introduction to Soergel bimodules" above), an extremely strange phenomenon that I would love to understand better.

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Presentation of right-angled Soergel categories by generators and relations

J. Pure Appl. Algebra 214 (2010), no. 12, 2265-2278. This was the first time that a "presentation of Soergel bimodules by generators and relations" was attempted. This revolutionary idea (explained to me by Rouquier) was the key of all the impressive subsequent development of the theory.

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My Ph.D thesis

Chapter 1 is essentially a version of the paper that one could call "Soergel bimodules explained by Soergel" with explanations of the obscure points. Sections 2.4 and 2.5 are original and are not included in any other paper. I give a different (and easier) proof of the fact that Rouquier complexes satisfy the braid relations.

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