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Leonardo Patimo

Publications and source records attributed to Leonardo Patimo.

17 recordsLinked to original sources
arXiv2025

A Hom formula for Soergel modules

We study Soergel modules for arbitrary Coxeter groups. For infinite Coxeter groups, we show that the homomorphisms between Soergel modules are in general more than those coming from morphisms of Soergel bimodules. This result provides a negative answer to a question posed by Soergel. We further show that the dimensions of the morphism spaces agree with the pairing in the Hecke algebra when Soergel modules are instead regarded as modules over the structure algebra. Moreover, we use this module structure to define a distinguished submodule of indecomposable Soergel bimodules that mimics the cohomology submodule of the intersection cohomology. Combined with the Hodge theory of Soergel bimodules, this can be used to extend results regarding the shape of Bruhat intervals, such as top-heaviness, to arbitrary Coxeter groups.

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arXiv2024

The charge statistic in type A via the Affine Grassmannian

We give a new construction of Lascoux-Sch\"uetzenberger's charge statistic in type A which is motivated by the geometric Satake equivalence. We obtain a new formula for the charge statistic in terms of modified crystal operators and an independent proof of this formula which does not rely on tableaux combinatorics.

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arXiv2024

The atomic Leibniz rule

The Demazure operator associated to a simple reflection satisfies the twisted Leibniz rule. In this paper we introduce a generalization of the twisted Leibniz rule for the Demazure operator associated to any atomic double coset. We prove that this atomic Leibniz rule is equivalent to a polynomial forcing property for singular Soergel bimodules.

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arXiv2024

On reduced expressions for core double cosets

The notion of a reduced expression for a double coset in a Coxeter group was introduced by Williamson, and recent work of Elias and Ko has made this theory more accessible and combinatorial. One result of Elias-Ko is that any coset admits a reduced expression which factors through a reduced expression for a related coset called its core. In this paper we define a class of cosets called atomic cosets, and prove that every core coset admits a reduced expression as a composition of atomic cosets. This leads to an algorithmic construction of a reduced expression for any coset. In types $A$ and $B$ we prove that the combinatorics of compositions of atomic cosets matches the combinatorics of ordinary expressions in a smaller group. In other types the combinatorics is new, as explored in a sequel by Ko.

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arXiv2024

Singular Light Leaves

For any Coxeter system we introduce the concept of singular light leaves, answering a question of Williamson raised in 2008. They provide a combinatorial basis for Hom spaces between singular Soergel bimodules.

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arXiv2023

Subexpressions and the Bruhat order for double cosets

The Bruhat order on a Coxeter group is often described by examining subexpressions of a reduced expression. We prove that an analogous description applies to the Bruhat order on double cosets. This establishes the compatibility of the Bruhat order on double cosets with concatenation, leading to compatibility between the monoidal structure and the ideal of lower terms in the singular Hecke 2-category. We also prove other fundamental properties of this ideal of lower terms.

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arXiv2023

Demazure operators for double cosets

For any Coxeter system, and any double coset for two standard parabolic subgroups, we introduce a Demazure operator. These operators form a basis for morphism spaces in a category we call the nilCoxeter category, and we also present this category by generators and relations. We prove a generalization to this context of Demazure's celebrated theorem on Frobenius extensions. This generalized theorem serves as a criterion for ensuring the proper behavior of singular Soergel bimodules.

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arXiv2023

Atoms and charge in type $C_2$

We construct atomic decompositions for crystals of type $C_{2}$ and define a charge statistic on them, thus providing positive combinatorial formulas for Kostka-Foulkes polynomials associated to them together with a natural geometric interpretation.

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arXiv2021

Charges via the Affine Grassmannian

We give a new construction of Lascoux-Sch\"utzenberger's charge statistic in type A which is motivated by the geometric Satake equivalence We obtain a new formula for the charge statistic in terms of modified crystal operators and an independent proof of this formula not relying on tableaux combinatorics.

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arXiv2021

Pre-canonical bases on affine Hecke algebras

For any affine Weyl group, we introduce the pre-canonical bases. They are a set of bases $\{\mathbf{N}^i\}_{1\leq i \leq m+1} $ (where $m$ is the height of the highest root) of the spherical Hecke algebra that interpolates between the standard basis $\mathbf{N}^1$ and the canonical basis $\mathbf{N}^{m+1}$. The expansion of $\mathbf{N}^{i+1}$ in terms of the $\mathbf{N}^i$ is in many cases very simple and we conjecture that in type $A$ it is positive.

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arXiv2020

On the affine Hecke category for $SL_3$

We study the diagrammatic Hecke category associated with the affine Weyl group of type $\tilde{A}_2$. More precisely we find a (surprisingly simple) basis for the Hom spaces between indecomposable objects, that we call indecomposable double leaves.

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arXiv2019

On the Induction of p-Cells

We study cells with respect to the $p$-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see arXiv:1510.01556, arXiv:1901.02323) and their compatibility with standard parabolic subgroups. We show that after induction to the surrounding bigger Coxeter group the cell module of a right $p$-cell in a standard parabolic subgroup decomposes as a direct sum of cell modules. Along the way, we state some new positivity properties of the $p$-canonical basis.

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arXiv2019

Bases of the Intersection Cohomology of Grassmannian Schubert Varieties

The parabolic Kazhdan-Lusztig polynomials for Grassmannians can be computed by counting Dyck partitions. We "lift" this combinatorial formula to the corresponding category of singular Soergel bimodules to obtain bases of the Hom spaces between indecomposable objects. In particular, we describe bases of intersection cohomology of Schubert varieties in Grassmannians parametrized by Dyck partitions which extend (after dualizing) the classical Schubert basis of the ordinary cohomology.

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arXiv2019

Singular Rouquier Complexes

We generalise the construction of Rouquier complexes to the setting of singular Soergel bimodules by taking minimal complexes of the restriction of Rouquier complexes. We show that they retain many of the properties of ordinary Rouquier complexes: they are delta-split, they satisfy a vanishing formula and, when Soergel's conjecture holds they are perverse. As an application, we use singular Rouquier complexes to establish Hodge theory for singular Soergel bimodules.

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arXiv2016

The N\'eron-Severi Lie Algebra of a Soergel Module

We introduce the N\'eron-Severi Lie algebra of a Soergel module and we determine it for a large class of Schubert varieties. This is achieved by investigating which Soergel modules admit a tensor decomposition. We also use the N\'eron-Severi Lie algebra to provide an easy proof of the well-known fact that a Schubert variety is rationally smooth if and only if its Betti numbers satisfy Poincar\'e duality.

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