arXiv subjects

Nicolle Gonzalez

Publications and source records attributed to Nicolle Gonzalez.

6 recordsLinked to original sources

arXiv2026

A whittled complex for the Khovanov homology of torus links

We give an algorithm for reducing the number of generators of the Khovanov chain complex of the torus braid $ft^k_n = (\sigma_1\sigma_2\cdots \sigma_{n-1})^k$ on $n$ strands by applying Bar-Natan Gaussian elimination along a distinguished set of Gaussian elimination isomorphisms. We call the resulting complex $\mathcal{FT}^k_n$ a \emph{whittled complex} for the Khovanov homology of torus braids. Using this algorithm, we provide a bound for the number of generators at a fixed homological degree in our whittled complex.

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arXiv2025

The elliptic Hall algebra and the double Dyck path algebra

We show that the positive half $\mathcal{E}_{q,t}^{>}$ of the elliptic Hall algebra is embedded as a natural spherical subalgebra inside the double Dyck path algebra $\mathbb{B}_{q,t}$ introduced by Carlsson, Mellit and the second author. For this, we use the Ding-Iohara-Miki presentation of the elliptic Hall algebra and identify the generators inside $\mathbb{B}_{q,t}$. In order to obtain the entire elliptic Hall algebra $\mathcal{E}_{q,t}$, we define a ``double'' $\mathbb{DB}_{q,t}$ of the double Dyck path algebra, together with its positive and negative subalgebras and an involution that exchanges them.

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arXiv2022

Extremal tensor products of Demazure crystals

Demazure crystals are subcrystals of highest weight irreducible $\mathfrak{g}$-crystals. In this article, we study tensor products of a larger class of subcrystals, called extremal, and give a local characterization for exactly when the tensor product of Demazure crystals is extremal. We then show that tensor products of Demazure crystals decompose into direct sums of Demazure crystals if and only if the tensor product is extremal, thus providing a sufficient and necessary local criterion for when the tensor product of Demazure crystals is itself Demazure. As an application, we show that the primary component in the tensor square of any Demazure crystal is always Demazure.

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arXiv2022

Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials

We give a crystal-theoretic proof that nonsymmetric Macdonald polynomials specialized to $t=0$ are affine Demazure characters. We explicitly construct an affine Demazure crystal on semistandard key tabloids such that removing the affine edges recovers the finite Demazure crystals constructed earlier by the authors. We also realize the filtration on highest weight modules by Demazure modules by defining explicit embedding operators which, at the level of characters, parallels the recursion operators of Knop and Sahi for specialized nonsymmetric Macdonald polynomials. Thus we prove combinatorially in type A that every affine Demazure module admits a finite Demazure flag.

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arXiv2019

Demazure crystals for specialized nonsymmetric Macdonald polynomials

We give an explicit, nonnegative formula for the expansion of nonsymmetric Macdonald polynomials specialized at $t=0$ in terms of Demazure characters. Our formula results from constructing Demazure crystals whose characters are the nonsymmetric Macdonald polynomials, which also gives a new proof that these specialized nonsymmetric Macdonald polynomials are positive graded sums of Demazure characters. Demazure crystals are certain truncations of classical crystals that give a combinatorial skeleton for Demazure modules. To prove our construction, we develop further properties of Demazure crystals, including an efficient algorithm for computing their characters from highest weight elements. As a corollary, we obtain a new formula for the Schur expansion of Hall--Littlewood polynomials in terms of a simple statistic on highest weight elements of our crystals.

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arXiv2018

Categorical Bernstein Operators and the Boson-Fermion Correspondence

We prove a conjecture of Cautis and Sussan providing a categorification of the Boson-Fermion correspondence as formulated by Frenkel and Kac. We lift the Bernstein operators to infinite chain complexes in Khovanov's Heisenberg category H and from them construct categorical analogues of the Kac-Frenkel fermionic vertex operators. These fermionic functors are then shown to satisfy categorical Clifford algebra relations, solving a conjecture of Cautis and Sussan. We also prove another conjecture of Cautis and Sussan demonstrating that the categorical Fock space representation of H is a direct summand of the regular representation by showing that certain infinite chain complexes are categorical Fock space idempotents. In the process, we enhance the graphical calculus of H by lifting various Littlewood-Richardson branching isomorphisms to the Karoubian envelope of H.

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