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José Simental

Publications and source records attributed to José Simental.

At least 19 recordsLinked to original sources
arXiv2026

Torus actions on compactified braid varieties and polytopality of subword complexes

Every cluster variety admits an action of its cluster dilation group. We prove that, in the case of braid varieties for simple Lie groups, this action always extends to a regular action on each of the brick compactifications. We explore two applications of this result. First, we show that any closed Richardson variety admits a faithful action of a torus of rank the Kazhdan-Lusztig $d$-invariant, answering affirmatively a recent question of E. Gorsky--S. Kim--M. Sherman-Bennett. The same result holds for projected Richardson varieties. Second, we show that the braid variety is a torus if and only if for each of its brick compactifications, the polar dual of the moment polytope for this action realizes the corresponding subword complex. The braid words satisfying this property turn out to be precisely the double root free words of V. Pilaud and C. Stump. This provides a novel approach to the longstanding open question of the polytopality of spherical subword complexes asked by A. Knutson and E. Miller, and in particular gives infinite families of subword complexes admitting polytopal realizations in dimension higher than the rank of the corresponding Coxeter group. As a common consequence of these two applications, we classify all Bruhat intervals in finite crystallographic Coxeter groups which are isomorphic to face lattices of convex polytopes via certain double root free words.

math.AG↗
arXiv2026

Cluster tori over $\mathbb{F}_2$, hexagonal moves on triangulations, and minimal coverings of cluster manifolds

We study cluster algebras over $\mathbb{F}_2$. By the Laurent phenomenon there is a map from the set of seeds of the cluster algebra to the corresponding cluster variety. We show that in type $A$, fibers of this map can be described in terms of certain edges of the universal polytope of triangulations of a polygon. Moreover, we show that there is a section of this map giving seeds whose corresponding cluster tori cover the cluster manifold over any field $\mathbb{F}$, but there are also sections giving seeds whose cluster tori do not cover the cluster manifold over any field $\mathbb{F} \not\cong \mathbb{F}_2$.

math.CO↗
arXiv2026

Mysterious points in keys but not trees

The deep locus of a cluster variety is defined to be the set of its points that do not belong to any cluster torus. We show that, if the cluster variety has a seed whose mutable part is a tree without multiple edges, then the deep locus can be characterized as the set of points whose stabilizer under a certain group action is nontrivial. Deep points without a stabilizer are called mysterious. We establish that many other classes of acyclic quivers (including keys) often have mysterious points. This refutes Conjecture 1.1 of arXiv:2402.16970, but establishes it in many important cases.

math.CO↗
arXiv2026

Smooth correspondences between quiver varieties

We introduce a new class of smooth correspondences between Nakajima quiver varieties called split parabolic quiver varieties, and study their properties. We use these correspondences to construct an explicit resolution of singularities of quiver Brill--Noether loci and prove that the latter are irreducible and Cohen-Macaulay of expected dimension (if non-empty). This generalizes the results of Nakajima--Yoshioka and Bayer--Chen--Jiang for Hilbert schemes of points on surfaces.

math.AG↗
arXiv2026

Positroid Links and Braid varieties

We study braid varieties and their relation to open positroid varieties. We discuss four different types of braids associated to open positroid strata and show that their associated Legendrian links are all Legendrian isotopic. In particular, we prove that each open positroid stratum can be presented as the augmentation variety for four different Legendrian fronts described in terms of either permutations, juggling patterns, cyclic rank matrices or Le diagrams. We also relate braid varieties to open Richardson varieties and brick manifolds, showing that the latter provide projective compactifications of braid varieties, with normal crossing divisors at infinity.

math.AG↗
arXiv2026

Bruhat intervals that are large hypercubes

We study the question of finding big Bruhat intervals that are poset hypercubes in the symmetric group $S_n$. Using permutations suggested by AlphaEvolve (an evolutionary coding agent developed by Google DeepMind), we were led to an unusual situation in which the agent produced a pattern which performed well for the $n$ tested, and which we show works well for general $n$. When $n$ is a power of 2 we exhibit a hypercube of dimension $O(n\log n)$, matching the largest possible dimension up to a constant multiple. Furthermore, we give an exact characterization of the vertices of this hypercube: they are precisely the \emph{dyadically well-distributed} permutations -- a simple digitwise property that already appeared in connection with Monte Carlo integration and mathematical finance. The maximal dimension of a Bruhat interval that is an hypercube in $S_n$ gives a lower bound (and possibly is equal to) the maximal possible coefficient of the second-highest degree term in the Kazhdan--Lusztig $R$-polynomial in $S_n$. As a surprising consequence, we obtain a new lower bound of order $n\log n$ for the maximal number of frozen variables appearing in the cluster algebras attached to the open Richardson varieties in $S_n$, and a similar result for moduli spaces of embeddings of Bruhat graphs.

math.CO↗
arXiv2025

Comparing cluster algebras on braid varieties

Braid varieties parametrize linear configurations of flags with transversality conditions dictated by positive braids. They include and generalize reduced double Bruhat cells, positroid varieties, open Bott-Samelson varieties, and Richardson varieties, among others. Recently, two cluster algebra structures were independently constructed in the coordinate rings of braid varieties: one using weaves and the other using Deodhar geometry. The main result of the article is that these two cluster algebras coincide. More generally, our comparative study matches the different concepts and results from each approach to the other, both on the combinatorial and algebraic geometric aspects.

math.AG↗
arXiv2025

Splicing braid varieties

For a positive braid $β\in \mathrm{Br}^{+}_{k}$, we consider the braid variety $X(β)$. We define a family of open sets $\mathcal{U}_{r, w}$ in $X(β)$, where $w \in S_k$ is a permutation and $r$ is a positive integer no greater than the length of $β$. For fixed $r$, the sets $\mathcal{U}_{r, w}$ form an open cover of $X(β)$. We conjecture that $\mathcal{U}_{r,w}$ is given by the nonvanishing of some cluster variables in a single cluster for the cluster structure on $\mathbb{C}[X(β)]$ and that $\mathcal{U}_{r,w}$ admits a cluster structure given by freezing these variables. Moreover, we show that $\mathcal{U}_{r, w}$ is always isomorphic to the product of two braid varieties, and we conjecture that this isomorphism is quasi-cluster. In some important special cases, we are able to prove our conjectures.

math.AG↗
arXiv2025

Splicing skew shaped positroids

Skew shaped positroids (or skew shaped positroid varieties) are certain Richardson varieties in the flag variety that admit a realization as explicit subvarieties of the Grassmannian $\mathrm{Gr}(k,n)$. They are parametrized by a pair of Young diagrams $μ\subseteq λ$ fitting inside a $k \times (n-k)$-rectangle. For every $a = 1, \dots, n-k$, we define an explicit open set $U_a$ inside the skew shaped positroid $S^{\circ}_{λ/μ}$, and show that $U_a$ is isomorphic to the product of two smaller skew shaped positroids. Moreover, $U_a$ admits a natural cluster structure and the aforementioned isomorphism is quasi-cluster in the sense of Fraser. Our methods depend on realizing the skew shaped positroid as an explicit braid variety, and generalize the work of the first and third authors for open positroid cells in the Grassmannian.

math.AG↗
arXiv2024

Algebraic Weaves and Braid Varieties

In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting stratifications of braid varieties and their quotients. It is shown that the maximal charts of these stratifications are exponential Darboux charts for the holomorphic symplectic structures, and we relate these strata to exact Lagrangian fillings of Legendrian links.

math.RT↗
arXiv2024

Cluster structures on braid varieties

We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig's coordinates. Several explicit seeds are provided and the quiver and cluster variables are readily computable. We prove that these upper cluster algebras equal their cluster algebras, show local acyclicity, and explicitly determine their DT-transformations as the twist automorphisms of braid varieties. The main result also resolves the conjecture of B. Leclerc on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties.

math.RT↗
arXiv2024

Cluster deep loci and mirror symmetry

Affine cluster varieties are covered up to codimension 2 by open algebraic tori. We put forth a general conjecture (based on earlier conversation between Vivek Shende and the last author) characterizing their deep locus, i.e. the complement of all cluster charts, as the locus of points with non-trivial stabilizer under the action of cluster automorphisms. We use the diagrammatics of Demazure weaves to verify the conjecture for skew-symmetric cluster varieties of finite cluster type with arbitrary choice of frozens and for the top open positroid strata of Grassmannians $\mathrm{Gr}(2,n)$ and $\mathrm{Gr}(3,n)$. We illustrate how this already has applications in symplectic topology and mirror symmetry, by proving that the Fukaya category of Grassmannians $\mathrm{Gr}(2,2n+1)$ is split-generated by finitely many Lagrangian tori, and homological mirror symmetry holds with a Landau--Ginzburg model proposed by Rietsch. Finally, we study the geometry of the deep locus, and find that it can be singular and have several irreducible components of different dimensions, but they all are again cluster varieties in our examples in really full rank cases.

math.AG↗
arXiv2024

From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra

In this note we explicitly construct an action of the rational Cherednik algebra $H_{1,m/n}(S_n,\mathbb{C}^n)$ corresponding to the permutation representation of $S_n$ on the $\mathbb{C}^{*}$-equivariant homology of parabolic Hilbert schemes of points on the plane curve singularity $\{x^{m} = y^{n}\}$ for coprime $m$ and $n$. We use this to construct actions of quantized Gieseker algebras on parabolic Hilbert schemes on the same plane curve singularity, and actions of the Cherednik algebra at $t = 0$ on the equivariant homology of parabolic Hilbert schemes on the non-reduced curve $\{y^{n} = 0\}.$ Our main tool is the study of the combinatorial representation theory of the rational Cherednik algebra via the subalgebra generated by Dunkl-Opdam elements.

math.RT↗
arXiv2023

Harish-Chandra bimodules for type A rational Cherednik algebras

We study Harish-Chandra bimodules for the rational Cherednik algebra associated to the symmetric group $S_{n}$. In particular, we show that for any parameter $c \in \mathbb{C}$, the category of Harish-Chandra $H_{c}$-bimodules admits a fully faithful embedding into the category $\mathcal{O}_{c}$, and describe the irreducibles in the image. We also construct a duality on the category of Harish-Chandra bimodules, and in fact we do this in a greater generality of quantizations of Nakajima quiver varieties. We use this duality, along with induction and restriction functors, to describe, as an abelian category, the smallest Serre subcategory of the category of Harish-Chandra bimodules containing the regular bimodule, as well as to explicitly describe the tensor products of its irreducible objects.

math.RT↗
arXiv2023

Calibrated representations of the double Dyck path algebra

The double Dyck path algebra $\mathbb{A}_{q,t}$ and its polynomial representation first arose as a key figure in the proof of the celebrated Shuffle Theorem of Carlsson and Mellit. A geometric formulation for an equivalent algebra $\mathbb{B}_{q,t}$ was then given by the second author and Carlsson and Mellit using the K-theory of parabolic flag Hilbert schemes. In this article, we initiate the systematic study of the representation theory of the double Dyck path algebra $\mathbb{B}_{q,t}$. We define a natural extension of this algebra and study its calibrated representations. We show that the polynomial representation is calibrated, and place it into a large family of calibrated representations constructed from posets satisfying certain conditions. We also define tensor products and duals of these representations, thus proving (under suitable conditions) the category of calibrated representations is generically monoidal. As an application, we prove that tensor powers of the polynomial representation can be constructed from the equivariant K-theory of parabolic Gieseker moduli spaces.

math.RT↗
arXiv2023

Higher rank $(q,t)$-Catalan polynomials, affine Springer fibers, and a finite Rational Shuffle Theorem

We introduce the higher rank $(q,t)$-Catalan polynomials and prove they equal truncations of the Hikita polynomial to a finite number of variables. Using affine compositions and a certain standardization map, we define a dinv statistic on rank $r$ semistandard $(m,n)$-parking functions and prove codinv counts the dimension of an affine space in an affine paving of a parabolic affine Springer fiber. Combining these results, we give a finite analogue of the Rational Shuffle Theorem in the context of double affine Hecke algebras. Lastly, we also give a Bizley-type formula for the higher rank Catalan numbers in the non-coprime case.

math.CO↗
arXiv2022

An elementary approach to the homological properties of constant-rank operators

We give a simple and constructive extension of Raiţă's result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement on the order of the operators constructed by Raiţă, as well as the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we establish a generalized Poincaré lemma for constant-rank operators and homogeneous spaces on $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.

math.AP↗