José Simental Rodríguez

Research

I am interested in Representation theory and its interactions with Algebraic geometry and Combinatorics. A big part of my research concerns the (geometric and combinatorial) representation theory of rational Cherednik and related Hecke algebras. You can see an introductory talk on rational Cherednik algebras I gave at ICMS Edinburgh in June 2019 here. More generally, I am interested in the representation theory of quantized quiver varieties. In 2015, I was teaching assistant to Ivan Losev in a Masterclass on Quantized quiver varieties given at QGM Aarhus, Denmark. Here are videos of the lectures. Lately, I have also been thinking of categorical link invariants, double affine Hecke algebras and quantum toroidal algebras.

Publications

Cluster structures on braid varieties (joint with R. Casals, E. Gorsky, M. Gorsky, I. Le and L. Shen) J. Amer. Math. Soc. 38 (2025), 369--479. arXiv. Video of a talk on this paper I gave at CMO, September 2022.
Calibrated representations of the double Dyck path algebra (joint with N. González and E. Gorsky). Math. Ann 391 (2025), 1505--1590
Algebraic weaves and braid varieties. (joint with R. Casals, E. Gorsky and M. Gorsky). Amer. J. Math. 146 (6) (2024), 1469--1576. arXiv. Video of a talk on this paper I gave at the RepNet Virtual Seminar, February 2021.
From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra. (joint wth E. Gorsky and M. Vazirani). Transf. Groups 29 (2024), 647--716 arXiv. Video of a talk on this paper I gave at the Northeastern GPRT seminar, October 2020.
Higher rank (q,t)-Catalan polynomials, affine Springer fibers and a finite shuffle theorem (joint with N. González and M. Vazirani). Trans. AMS 377 (7) (2024), 5087--5127. arXiv. Extended Abstract in Sem. Loth. de Comb. 89B (2023), Article #84
Unitary representations of cyclotomic Hecke algebras at roots of unity: combinatorial classification and BGG resolutions. (joint with C. Bowman and E. Norton) J. Inst. Mat. Jussieu 23 (2), 2024, 557--608.
Harish-Chandra bimodules for type A rational Cherednik algebras. J. Algebra 642 (2024), 286--329. arXiv
Algebra and geometry of link homology. (joint with E. Gorsky and O. Kivinen) Bull. London Math. Soc. 55 (2) (2023) 537--591. These are lecture notes for a course E. Gorsky gave at IHÉS in July of 2021.
An elementary approach to the homological properties of constant-rank operators. (joint with A. Arroyo-Rabasa). C. R. Math. Acad. Sci. Paris 361 (2023), 45--63
On BGG resolutions of unitary modules for quiver Hecke and Cherednik algebras. (joint with C. Bowman and E. Norton) Selecta Math New Ser. 28, 29 (2022)
Representations with minimal support for quantized Gieseker varieties. (joint with P. Etingof, V. Krylov and I. Losev) Mathematische Zeitschrift 298 (3) (2021) 1593--1621 arXiv. Slides from a talk on this (and other) papers I gave at the MIT Infinite-dimensional algebra seminar, February 2021.
On Harish-Chandra bimodules over rational Cherednik algebras. Advances in Mathematics 317 (2017) 299--349. arXiv
An alternative perspective on projectivity of modules. (joint with C. Holston, S. López-Permouth and J. Mastromatteo) Glasgow Math. Journal 57 issue 1 (2015) 83--99. arXiv
Characterizing rings in terms of the extent of injectivity and projectivity of their modules. (joint with S. López-Permouth). J. Algebra 362 (2012) 56--69. arXiv

Preprints

Splicing braid varieties (joint with E. Gorsky, S. Kim and T. Scroggin) arXiv
Splicing skew shaped positroids (joint with E. Gorsky, S. Kim and T. Scroggin) arXiv
The elliptic Hall algebra and the double Dyck path algebra (joint with N. González and E. Gorsky) arXiv
Cluster deep loci and mirror symmetry (joint with M. Castronovo, M. Gorsky and D. E. Speyer) arXiv.
Positroid links and braid varieties. (joint with R. Casals, E. Gorsky and M. Gorsky). arXiv.

Other

Clusters and weaves (joint with M. Gorsky). Notices of the AMS 71 (8), 1004--1013
Review of Mexican Mathematicians in the world: trends and recent contributions. (joint with A. Arroyo-Rabasa). Notices of the AMS 70 (8) 1274--1277