Pedro Vaz

• PhD
• Catholic University of Louvain

We investigate the structure of Kazhdan-Lusztig polynomials of the symmetric group by leveraging computational approaches from big data, including exploratory and topological data analysis, applied to the polynomials for symmetric groups of up to 11 strands.

We construct a 2-functor from the Kac-Moody 2-category for the extended quantum affine sl(3) to the homotopy 2-category of bounded chain complexes with values in the Kac-Moody 2-category for quantum gl(3), categorifying the evaluation map between the corresponding quantum Kac-Moody algebras.

Generalizing the dihedral picture for G(M,M,2), we construct Hecke algebras (and categories) and asymptotic counterparts. We think of these as associated with the complex reflection group G(M,M,N).

We study the geometry and topology of $\Delta$-Springer varieties associated with two-row partitions. These varieties were introduced in recent work by Griffin-Levinson-Woo to give a geometric realization of a symmetric function appearing in the Delta conjecture by Haglund-Remmel-Wilson. We provide an explicit and combinatorial description of the i...

We discuss formulas for the asymptotic growth rate of the number of summands in tensor powers in certain (finite or infinite) monoidal categories. Our focus is on monoidal categories with infinitely many indecomposable objects, with our main tools being generalized Perron-Frobenius theory alongside techniques from random walks.

We consider the odd analogue of the category of Soergel bimodules. In the odd case and already for two variables, the transposition bimodule cannot be merged into the generating Soergel bimodule, forcing one into a monoidal category with a larger Grothendieck ring compared to the even case. We establish biadjointness of suitable functors and develo...

We give explicit formulas for the asymptotic growth rate of the number of summands in tensor powers in certain monoidal categories with finitely many indecomposable objects, and related structures.

For any Levi subalgebra of the form \mathfrak{l}=\mathfrak{gl}_{l_{1}}\oplus\cdots\oplus\mathfrak{gl}_{l_{d}}\subseteq\mathfrak{gl}_{n} we construct a quotient of the category of annular quantum \mathfrak{gl}_{n} webs that is equivalent to the category of finite-dimensional representations of quantum \mathfrak{l} generated by exterior powers of the...

We define a derivation on the enhanced nilHecke algebra yielding a p-dg algebra when working over a field of characteristic p. We define functors on the category of p-dg modules resulting in an action of small quantum sl2 on the Grothendieck group, which is isomorphic to a baby Verma module. We upgrade the derivation into an action of the Lie algeb...

We give explicit formulas for the asymptotic growth rate of the number of summands in tensor powers in certain monoidal categories with finitely many indecomposable objects, and related structures.

We give a closed formula to evaluate exterior webs (also called MOY webs) and the associated Reshetikhin-Turaev link polynomials.

We establish a version of quantum Howe duality with two general linear quantum enveloping algebras that involves a tensor product of Verma modules. We prove that the (colored higher) LKB representations arise from this duality and use this description to show that they are simple as modules for various subgroups of the braid group, including the pu...

In this paper, we use Soergel calculus to define a monoidal functor, called the evaluation functor, from extended affine type A Soergel bimodules to the homotopy category of bounded complexes in finite type A Soergel bimodules. This functor categorifies the well-known evaluation homomorphism from the extended affine type A Hecke algebra to the fini...

In this paper, we study handlebody versions of some classical diagram algebras, most prominently, handlebody versions of Temperley–Lieb, blob, Brauer, BMW, Hecke and Ariki–Koike algebras. Moreover, motivated by Green–Kazhdan–Lusztig’s theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this ref...

We consider the odd analogue of the category of Soergel bimodules. In the odd case and already for two variables, the transposition bimodule cannot be merged into the generating Soergel bimodule, forcing one into a monoidal category with a larger Grothendieck ring compared to the even case. We establish biadjointness of suitable functors and develo...

For any Levi subalgebra of the form $\mathfrak{l}=\mathfrak{gl}_{l_{1}}\oplus\dots\oplus\mathfrak{gl}_{l_{d}}\subseteq\mathfrak{gl}_{n}$ we construct a quotient of the category of annular quantum $\mathfrak{gl}_{n}$ webs that is equivalent to the category of finite dimensional representations of quantum $\mathfrak{l}$ generated by exterior powers o...

We construct a categorification of parabolic Verma modules for symmetrizable Kac–Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also...

We explain how Queffelec–Sartori’s construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for gl2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-...

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product of a universal \mathfrak{sl}_2 Verma module and several integrable irreducible modules. When the integrable modules are two-dimensional, we construct a categorical action of the blob algebra on derived categories of these dg-algebras which intertwines the categorica...

In this paper we study handlebody versions of classical diagram algebras, most prominently, handlebody versions of Temperley-Lieb, blob, Brauer/BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulat...

We construct a dg-enhancement of Webster's tensor product algebras that categorifies the tensor product of a universal sl2 Verma module and several integrable irreducible modules. We show that the blob algebra acts via endofunctors on derived categories of such dg-enhanced algebras in the case when the integrable modules are two-dimensional. This a...

We prove a Schur-Weyl duality between the quantum enveloping algebra of $\mathfrak{gl}_m$ and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several algebraically independent parameters and is realized through the tensor product of a parabolic universal Verma module and a tensor power of the natura...

We construct a supercategory that can be seen as a skew version of (thickened) KLR algebras for the type $A$ quiver. We use our supercategory to construct homological invariants of tangles and show that for every link our invariant gives a link homology theory supercategorifying the Jones polynomial. Our homology is distinct from even Khovanov homo...

We provide an introduction to the higher representation theory of Kac–Moody algebras and categorification of Verma modules.

We construct DG-enhanced versions of the degenerate affine Hecke algebra and of the affine $q$-Hecke algebra. We extend Brundan-Kleshchev and Rouquier's isomorphism and prove that after completion DG-enhanced versions of Hecke algebras are isomorphic to completed DG-enhanced versions of KLR algebras for suitably defined quivers. As a byproduct, we...

We construct a categorification of (parabolic) Verma modules for symmetrizable Kac-Moody algebras using KLR-like diagrammatic algebras.

We study the superalgebra $A_n$ introduced by the authors in a previous paper and use it to construct a 2-Verma module for quantum $\mathfrak{sl}_2$. We prove a uniqueness result about 2-Verma modules on $\Bbbk$-linear 2-categories.

In this paper we study the superalgebra $A_n$, introduced by the authors in previous work on categorification of Verma modules for quantum $\mathfrak{sl}_2$. The superalgebra $A_n$ is akin to the nilHecke algebra, and shares similar properties. In particular, we prove a uniqueness result about 2-Verma modules on $\Bbbk$-linear 2-categories.

We construct an odd version of Khovanov's arc algebra $H^n$. Extending the center to elements that anticommute, we get a subalgebra that is isomorphic to the oddification of the cohomology of the $(n,n)$-Springer varieties. We also prove that the odd arc algebra can be twisted into an associative algebra.

We construct an odd version of Khovanov's arc algebra $H^n$. Extending the center to elements that anticommute, we get a subalgebra that is isomorphic to the oddification of the cohomology of the $(n,n)$-Springer varieties. We also prove that the odd arc algebra can be twisted into an associative algebra.

We give a geometric categorification of the Verma modules $M(\lambda)$ for quantum $\mathfrak{sl}_2$.

We give a geometric categorification of the Verma modules $M(\lambda)$ for quantum $\mathfrak{sl}_2$.

We use super q-Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of glN-modules (and, more generally, gl(N|M)-modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an application, we give a representation theoretic explanation a...

We construct functors categorifying the branching rules for $U_q(\mathfrak{g})$ for $\mathfrak{g}$ of type $B_n$, $C_n$, and $D_n$ for the embeddings $so_{2n+1}\supset so_{2n-1}$, $sp_{2n}\supset sp_{2n-2}$, and $so_{2n}\supset so_{2n-2}$. We give the corresponding categorical Gelfand-Tsetlin basis.

This note is a write-up of a talk given by the author at the Meeting of the Sociedade Portuguesa de Matematica in July 2012. We describe Jaeger's HOMFLY-PT expansion of the Kauffman polynomial and how to generalize it to other quantum invariants using the so-called "branching rules" for Lie algebra representations. We present a program which aims t...

We define a quotient of the category of finitely generated modules over the cyclotomic Khovanov-Lauda-Rouquier algebra for type An and show it has a module category structure over a direct sum of certain cyclotomic Khovanov-Lauda-Rouquier algebras of type An-1, this way categorifying the branching rules for the inclusion of sl(n) in sl(n+1). We sho...

We prove that the 2-variable BMW algebra embeds into an algebra constructed from the HOMFLY-PT polynomial. We also prove that the so(2N)-BMW algebra embeds in the q-Schur algebra of type A. We use these results to construct categorifications of the so(2N)-BMW algebra.

In this paper we categorify the q-Schur algebra \mathbf{S}_q(n,d) as a quotient of Khovanov and Lauda’s diagrammatic 2-category \mathcal{U}(\mathfrak{sl}_n) (Khovanov and Lauda 2010). We also show that our 2-category contains Soergel’s (1992) monoidal category of bimodules of type A , which categorifies the Hecke algebra H_q(d) , as a full sub-2-ca...

For each N≥4, we define a monoidal functor from Elias and Khovanov's diagrammatic version of Soergel's category of bimodules to the category of sl(N) foams defined by Mackaay, Stošić, and Vaz. We show that through these functors Soergel's category can be obtained from the sl(N) foams.

We define two functors from Elias and Khovanov's diagrammatic Soergel category, one targeting Clark-Morrison-Walker's category of disoriented sl(2) cobordisms and the other targeting the category of (universal) sl(3) foams.

For each N > 3, we define a monoidal functor from Elias and Khovanov's diagrammatic version of Soergel's category of bimodules to the category of sl(N) foams defined by Mackaay, Stosic and Vaz. We show that through these functors Soergel's category can be obtained from the sl(N) foams. Comment: v2, minor corrections, 17 pages, lots of figures

We define two functors from Elias and Khovanov's diagrammatic Soergel category, one targeting Clark-Morrison-Walker's category of disoriented sl(2) cobordisms and the other the category of (universal) sl(3) foams. Comment: v4, minor changes, referee's comments implemented. v3, 20 pages, lots of figures, remark about the general case rewritten, one...

We use foams to give a topological construction of a rational link homology categorifying the sl(N) link invariant, for N ≥ 4. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky [7]. We show that for any link our homology is isomorphic to the Khovanov-Rozansky [6] homology. 1.

In this paper we compute the reduced HOMFLY-PT homologies of the Conway and the Kinoshita-Terasaka knots and show that they are isomorphic.

In this paper we define the 1,2-coloured HOMFLY-PT link homology and prove that it is a link invariant. We conjecture that this homology categorifies the coloured HOMFLY-PT polynomial for links whose components are labelled 1 or 2.

In this thesis we define and study a categorification of the sl(N)-link polynomial using foams, for N\geq 3. For N=3 we define the universal sl(3)-link homology, using foams, which depends on three parameters and show that it is functorial, up to scalars, with respect to link cobordisms. Our theory is integral. We show that tensoring it with Q yiel...

We prove that the universal rational sl3 link homologies which were constructed by Khovanov in [sl(3) link homology, Algebr Geom Topol 4 (2004) 1045-1081] and by the authors in [The universal sl3 link homology, Algebr Geom Topol 7 (2007) 1135 -1169], using foams, and by Khovanov and Rozansky in [Virtual crossings, convolutions and a categorificatio...

We define the universal sl3-link homology, which depends on 3 parameters, following Khovanov's approach with foams. We show that this 3-parameter link homology, when taken with complex coefficients, can be divided into 3 isomorphism classes. The first class is the one to which Khovanov's original sl3-link homology belongs, the second is the one stu...

We show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.

This notes intended to supplement a talk given by the author at the Institut de Mathématiques de Jussieu -Université Paris 7 in January 2010. The purpose of the talk was to give an explanation of the use of Kapustin-Li formula in the evaluation of the closed foams used in the construction of the sl(N)-link homology. This notes contains material fro...