All of my articles are also available on my arXiv author page.
Publications
- 15. The atomic Leibniz rule (joint with Ben Elias, Hankyung Ko and Nicolás Libedinsky)
Preprint
The atomic Leibniz rule is a generalization to atomic cosets of the usual Leibniz rule for Demazure operators. We show that it is equivalent to the polynomial forcing relation on singular Soergel bimodules, which connects left and right publication up to lower terms. We show how one can deduce the atomic Leibniz rule from Williamson's theory of singular Soergel bimodules. We also show in type A directly the atomic Leibniz rule over Z, thus extending polynomial forcing to an important setting which was not previously covered in Williamson's theory. - 14. On reduced expressions for core double cosets (joint with Ben Elias, Hankyung Ko and Nicolás Libedinsky)
Preprint
Core double cosets are double cosets with the largest possible redundancy. Any given double coset has a reduced expression (or rex) factoring through a core coset. These rexes are key in the algorithm for constructing the singular light leaves, because they allow the placement of polynomials invariant under the redundancy subgroups. We show that each core coset admits a special rex which is a composition of atomic cosets. In particular, this provides an effective algorithm for constructing rexes for core cosets. We further show that in types A and B, atomic rexes behave exactly ordinary rexes for smaller groups, e.g. all atomic rexes are connected by atomic braid moves. - 13. Singular light leaves (joint with Ben Elias, Hankyung Ko and Nicolás Libedinsky)
Preprint
We construct a basis of morphisms between singular Soergel bimodules for any Coxeter group. This basis of morphisms, called the singular light leaves, is described in terms of diagrams and constitutes a substantial generalization of light leaves defined by Libedinsky and Elias-Williamson. Having such a basis permits tight control of the space of morphisms and is thus an important step towards establishing a complete diagrammatic description of the singular Hecke category. - 12. Subexpressions and the Bruhat order for double cosets (joint with Ben Elias, Hankyung Ko and Nicolás Libedinsky)
Preprint
W give a description of the Bruhat order on double coset in terms of singular reduced expression, which is a natural generalization of the Bruhat order on Coxeter groups. This result, which is combinatorial in nature, ensures that the compatibility between the monoidal structure and the ideal of lower terms in the singular Hecke 2-category. - 11. Demazure operators for double cosets (joint with Ben Elias, Hankyung Ko and Nicolás Libedinsky)
Preprint
This is the first paper in a project with Elias, Ko and Libedinsky, whose final goal is to give a diagrammatic presentation of singular Soergel bimodules. In this paper we define Demazure operators for double coset, show that these operator form a category which is the algebroid version of the nilHecke algebra, and prove a statement about the existence of special basis in Frobenius extension, generalizing a classical result of Demazure. This result ensures a proper behaviour of singular Soergel bimodules. - 10. Atoms and charge in type $C_2$ (joint with Jacinta Torres)
Preprint
We extend the methods and result of "Charges via the affine Grassmannian" to type $C_2$. In particular, we construct atomic decompositions for crystals of type $C_2$ and use this to define a charge statistic on them. This is the first case of a positive combinatorial formulas for Kostka-Foulkes polynomials outside type $A$. - 9. Charges via the affine Grassmannian
Preprint
We give a new construction of Lascoux-Schützenberger’s charge statistic in type $A$ motivated by the geometry of the affine Grassmannian. Using the geometric Satake equivalence, we construct the charge by understanding how hyperbolic localization changes on a family of cocharacters in the affine Grassmannian. Although this framework works for any reductive group, in type A there are two crucial features that make this procedure behave well: the atomic decomposition of crystal, which we revise here, and a technical property of twisted Bruhat graphs.
Slides from a talk in Bonn - 8. Pre-canonical Bases on affine Hecke algebras (joint with Nicolás Libedinsky and David Plaza)
Adv. Math., 399 (2022)
We introduce some new bases of the spherical Hecke algebras of affine Weyl groups, called the pre-canonical bases, which interpolate between the standard and KL bases. We conjecture that in type A they have remarkable positive properties, namely that the (i+1)-th pre-canonical basis has positive coefficients in the $i$th basis. This would reduce the problem of finding KL polynomials in a sequence of easier problems. We also compute the pre-canonical bases explicitly up to rank 4.
- 7. On the Affine Hecke Category for $SL_3$ (joint with Nicolás Libedinsky)
Selecta Math.
We study the Hecke category for the affine Weyl group $\tilde{A}_2$ in characteristic 0. We determine the Kazhdan-Lusztig basis and give a recursive formula to compute the indempotent corresponding to every indecomposable object. Then we produce a basis of the morphisms between indecomposable objects, that we call "indecomposable light leaves." As a corollary, we also obtain a combinatorial objects counting KL polynomials
Slides from a talk at the RepNet Seminar - 6. On the Induction of p-Cells (joint with Thorge Jensen)
Transform. Groups
We generalize to p-cells (i.e. cells constructed using the p-canonical basis) a result of Geck, called parabolic induction of cells. In terms of cell modules, this translates to the fact that cell modulex of a right p-cell in a standard parabolic subgroup decompose as a direct sum of cell modules after induction to the bigger Coxeter group. A crucial point in the proof is a new positivity result of the p-canonical basis with respect of the hybrid p-canonical basis, which we introduce here.
Notes from a talk in Sydney - 5. Bases of the Intersection Cohomology of Grassmannian Schubert Varieties
J. Algebra 589 (2022), 345–400
There are several explicit combinatorial formula that describe Kazhdan-Lusztig polynomials for Grasmannians, one of which, described by Shigechi and Zinn-Justin, involves counting certain Dyck partitions. We ''lift'' this combinatorial formula to the intersection cohomology of Schubert varieties in Grassmannians and as a consequence obtain many distinguished bases of the intersection cohomology which extend the classical Schubert basis of the ordinary cohomology. These bases are an useful tool when studying the intersection cohomology of Schubert varieties as a module: for example, we can write in these bases a generalization of the Pieri's formula
Slides from a talk in Mooloolaba - 4. Singular Rouquier Complexes
Proc. Lon. Math. Soc., 125(6), 2022.
This is a revised version of Chapter 4 of my PhD thesis. Rouquier complexes are an important tool for studying Soergel bimodules and the Hecke category. Here we generalise the construction of Rouquier complexes to the setting of singular Soergel bimodules. We show that they retain many of the properties of ordinary Rouquier complexes: they are ∆ -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. - 3.A combinatorial formula for the coefficient of q of Kazhdan-Lusztig polynomials
Int. Math. Res. Not. IMRN 2021, no. 5
We study the coefficient of q in Kazhdan-Lusztig polynomials. Using moment graphs, for finite groups of type ADE we prove that this coefficient can be computed via a formula which only depends on the poset structure of the Bruhat interval.
Notes from a talk at MSRI - 2. The Néron-Severi Lie Algebra of a Soergel module
Transform. Groups 23 (2018), no. 4
Following Looijenga and Lunts, we introduce for any Soergel module a semisimple Lie algebra generated by all the Lefschetz operators and their adjoints. This algebra, called the Néron-Severi Lie algebra, can be used to provide a simple proof of the well-known fact that a Schubert variety is rationally smooth if and only if its Betti numbers satisfy Poincaré duality. We also determine a large set of elements, for finite Coxeter group, where the Néron-Severi Lie algebra coincides with the full Lie algebra of endomorphism preserving the intersection form.
Notes from a talk in Freiburg - 1. The Hard Lefschetz Theorem in positive characteristic for the Flag Varieties
Int. Math. Res. Not. IMRN 2018, no. 18
We show that the Hard Lefschetz theorem holds for the flag variety of some reductive group in characteristic p, if p is larger than the number of positive roots. The converse also holds, a part from 3 exceptional cases. One can also deduce an algebraic proof of the Hard Lefschetz theorem in the already known characteristic 0 case.
Notes from a talk in Kyoto
Other Mat(h)erials
- My PhD Thesis: Hodge theoretic aspects of Soergel bimodules and representation theory
The thesis includes the results my first two papers. There are also two sections which are not included elsewhere.- We discuss Soergel modules, and we show that for infinite Coxeter groups in general Soergel modules have "more" morphisms than Soergel bimodules, and in particular indecomposable Soergel bimodules can give rise to decomposable Soergel modules. We also show that one retrieves the usual Hom formula by regarding Soergel modules as modules over the nil Hecke algebra.
- We generalize the results of The Hodge Theory of Soergel Bimodules (B. Elias, G. Williamson): we show that the hard Lefschetz theorem and the Hodge-Riemann bilinear relations also hold for singular Soergel modules.
- My Posters (displayed at BIGS annual Poster Exhibition): 2015, 2016