## On reduced expressions for core double cosets

Joint with Ben Elias, Hankyung Ko and Leonardo Patimo. We produce an algorithmic construction of a reduced expression for any double coset.

Skip to content
# Papers and preprints

##
On reduced expressions for core double cosets

##
Singular light leaves

##
On the size of Bruhat intervals

##
Subexpressions and the Bruhat order for double cosets

##
Demazure operators for double cosets

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

##
Introsurvey of Representation Theory

##
Combinatorial invariance conjecture for affine A2

##
Pre-canonical bases on affine Hecke algebras

##
The Anti-Spherical Category

##
Kazhdan-Lusztig polynomials and subexpressions

##
Blob algebra approach to modular representation theory

##
p-Jones Wenzl idempotent

##
Gentle introduction to Soergel bimodules I: The basics

##
A non-perverse Soergel bimodule in type A

##
Indecomposable Soergel bimodules for Universal Coxeter groups

##
Light leaves and Lusztig’s conjecture

##
Standard objects in 2-braid groups

##
New bases of some Hecke algebras via Soergel bimodules

##
Presentation of right-angled Soergel categories by generators and relations

Joint with Ben Elias, Hankyung Ko and Leonardo Patimo. We produce an algorithmic construction of a reduced expression for any double coset.

Comments Off on On reduced expressions for core double cosets

February 26, 2024

Joint with Ben Elias, Hankyung Ko and Leonardo Patimo. For any Coxeter system we introduce the concept of singular light leaves (finally!!). Really fun stuff.

Comments Off on Singular light leaves

January 24, 2024

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.

Comments Off on On the size of Bruhat intervals

September 2, 2023

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.

Comments Off on Subexpressions and the Bruhat order for double cosets

September 2, 2023

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.

Comments Off on Demazure operators for double cosets

September 2, 2023

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**.

Comments Off on On the affine Hecke category for SL(3)

July 1, 2020

Accepted for publication in **Journal of the Indian Institute of Science**. It is an introduction/survey of representation theory with lots of humor, open questions, and extremely smart insights :)

Comments Off on Introsurvey of Representation Theory

March 10, 2022

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.

Comments Off on Combinatorial invariance conjecture for affine A2

May 29, 2021

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.

Comments Off on Pre-canonical bases on affine Hecke algebras

July 10, 2020

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**.

Comments Off on The Anti-Spherical Category

November 11, 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.

Comments Off on Kazhdan-Lusztig polynomials and subexpressions

May 2, 2020

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...

Comments Off on Blob algebra approach to modular representation theory

July 24, 2018

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.

Comments Off on p-Jones Wenzl idempotent

February 16, 2019

** 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**.

Comments Off on Gentle introduction to Soergel bimodules I: The basics

September 1, 2017

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.

Comments Off on A non-perverse Soergel bimodule in type A

January 1, 2016

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.

Comments Off on Indecomposable Soergel bimodules for Universal Coxeter groups

June 1, 2017

**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

Comments Off on Light leaves and Lusztig’s conjecture

January 1, 2015

Joint with Geordie Williamson, **Proceedings of the London Mathematical Society** 109 (2014), no. 5, 1264-1280.
We introduce the concept of **Δ-exact complexes**
for any Coxeter system. With that, 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

Comments Off on Standard objects in 2-braid groups

January 1, 2014

**Advances in Mathematics** 228 (2011) 1043-1067.
I introduce a **new set of bases ** for Hecke algebras related to extra-large Coxeter groups, coming from the theory of Soergel bimodules. I believe that they have a deep meaning related to the Hecke category and the p-canonical basis. This is related to my "Forking Path conjecture" that is false as stated, as proved by Gonzalo Jimenez. Nonetheless, in all generality, there seems to be an extremely strange phenomenon that I would love to understand better.

Comments Off on New bases of some Hecke algebras via Soergel bimodules

June 29, 2011

**Journal of Pure and Applied 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.

Comments Off on Presentation of right-angled Soergel categories by generators and relations

January 27, 2009