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Oded Yacobi

Publications and source records attributed to Oded Yacobi.

At least 19 recordsLinked to original sources
arXiv2026

Generating Hadamard matrices with transformers

We present a new method for constructing Hadamard matrices that combines transformer neural networks with local search in the PatternBoost framework. Our approach is designed for extremely sparse combinatorial search problems and is particularly effective for Hadamard matrices of Goethals--Seidel type, where Fourier methods permit fast scoring and optimisation. For orders between 100 and 200, it produces large numbers of inequivalent Hadamard matrices, and for larger orders, it succeeds where local search from random initialisation fails. The largest example found by our method has order 252. In addition to these new constructions, our experiments reveal that the transformer can discover and exploit useful hidden symmetry in the search space.

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arXiv2025

Normalisers of parabolic subgroups of Artin--Tits groups and Tits cone intersections

Let $\Gamma$ be a Coxeter diagram and let $J \subseteq \Gamma$. Motivated by 3-fold flops, Iyama and Wemyss study the hyperplane arrangement in the Tits cone intersection of $J$, which is a $J$-relative generalisation of the classical Coxeter arrangement. For $\Gamma$ of finite-type, we show that its complexified hyperplane complement is a $K(\pi,1)$ space for the normaliser (quotient) of the standard parabolic subgroup of the Artin--Tits group attached to $J$. For general $\Gamma$ we show that Brink--Howlett's groupoid, which describes normalisers of parabolic subgroups of Coxeter groups, has its universal cover described by the wall-and-chamber structure of the Tits cone intersection. We use this to show that wall crossing sequences satisfy an "atomic Matsumoto relation", generalising a theorem of Ko and answering questions raised by Iyama and Wemyss.

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arXiv2025

Semisimplifying categorical Heisenberg actions and periodic equivalences

We systematically apply semisimplification functors in modular representation theory. Motivated by the Duflo--Serganova functor in Lie superalgebras, we construct various functors of interest. In the setting of finite groups, we refine the cyclic group Brauer construction and categorify the Glauberman correspondence. In the setting of degenerate categorical Heisenberg actions, we obtain a rich collection of functors which commute with the categorical action. Applied to well-known categorifications of the basic representation and Fock space, our functors give explicit realizations of periodic equivalences for polynomial functors and symmetric groups first studied by Henke-Koenig. This allows us to globalize the equivalences of Henke-Koenig by symmetric monoidal functors. We apply these results to deduce branching properties of certain modular representations of $S_n$.

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arXiv2025

Periodic elements in finite type Artin-Tits groups and stability conditions

Periodic elements in finite type Artin--Tits groups are elements some positive power of which is central. We give a dynamical characterisation of periodic elements via their action on the corresponding 2-Calabi--Yau category and on its space of (fusion equivariant) Bridgeland stability conditions. The main theorem is that an element $\beta$ is periodic if and only if $\beta$ has a fixed point in the stability manifold.

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arXiv2023

Lie algebra actions on module categories for truncated shifted Yangians

We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's induction and restriction functors for Cherednik algebras, but their definition uses different tools. After this general definition, we focus on quiver gauge theories attached to a quiver $Γ$. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak{g}_Γ$ on category $ \mathcal O $ for these Coulomb branch algebras. When $ Γ$ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence. To establish this categorical action, we define a new class of "flavoured" KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.

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arXiv2023

4-Strand Burau is Unfaithful Modulo 5

We introduce a new algorithm for finding kernel elements in the Burau representation. Our algorithm applies reservoir sampling to a statistic on matrices which is closely correlated with Garside length. Using this we exhibit an explicit kernel element in the Burau representation on 4-strands reduced modulo 5.

math.RT↗
arXiv2023

Categorical braid group actions and cactus groups

Let $\mathfrak{g}$ be a semisimple simply-laced Lie algebra of finite type. Let $\mathcal{C}$ be an abelian categorical representation of the quantum group $U_q(\mathfrak{g})$ categorifying an integrable representation $V$. The Artin braid group $B$ of $\mathfrak{g}$ acts on $D^b(\mathcal{C})$ by Rickard complexes, providing a triangulated equivalence $Θ_{w_0}:D^b(\mathcal{C}_μ) \to D^b(\mathcal{C}_{w_0(μ)})$, where $μ$ is a weight of $V$ and $Θ_{w_0}$ is a positive lift of the longest element of the Weyl group. We prove that this equivalence is t-exact up to shift when $V$ is isotypic, generalising a fundamental result of Chuang and Rouquier in the case $\mathfrak{g}=\mathfrak{sl}_2$. For general $V$, we prove that $Θ_{w_0}$ is a perverse equivalence with respect to a Jordan-Hölder filtration of $\mathcal{C}$. Using these results we construct, from the action of $B$ on $V$, an action of the cactus group on the crystal of $V$. This recovers the cactus group action on $V$ defined via generalised Schützenberger involutions, and provides a new connection between categorical representation theory and crystal bases. We also use these results to give new proofs of theorems of Berenstein-Zelevinsky, Rhoades, and Stembridge regarding the action of symmetric group on the Kazhdan-Lusztig basis of its Specht modules.

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arXiv2023

On the action of the Weyl group on canonical bases

We study representations of simply-laced Weyl groups which are equipped with canonical bases. Our main result is that for a large class of representations, the separable elements of the Weyl group $W$ act on these canonical bases by bijections up to lower-order terms. Examples of this phenomenon include the action of separable permutations on the Kazhdan--Lusztig basis of irreducible representations for the symmetric group, and the action of separable elements of $W$ on dual canonical bases of weight zero in tensor product representations of a Lie algebra. Our methods arise from categorical representation theory, and in particular the study of the perversity of Rickard complexes acting on triangulated categories.

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arXiv2023

On Schützenberger modules of the cactus group

The cactus group acts on the set of standard Young tableau of a given shape by (partial) Schützenberger involutions. It is natural to extend this action to the corresponding Specht module by identifying standard Young tableau with the Kazhdan-Lusztig basis. We term these representations of the cactus group "Schützenberger modules", denoted $S^λ_{\mathsf{Sch}}$, and in this paper we investigate their decomposition into irreducible components. We prove that when $λ$ is a hook shape, the cactus group action on $S^λ_{\mathsf{Sch}}$ factors through $S_{n-1}$ and the resulting multiplicities are given by Kostka coefficients. Our proof relies on results of Berenstein and Kirillov and Chmutov, Glick, and Pylyavskyy.

math.CO↗
arXiv2022

On the action of the long cycle on the Kazhdan-Lusztig basis

The complex irreducible representations of the symmetric group carry an important canonical basis called the Kazhdan-Lusztig basis. Although it is difficult to express how general permutations act on this basis, some distinguished permutations have beautiful descriptions. In 2010 Rhoades showed that the long cycle $(1, 2,..., n)$ acts by the jeu-de-taquin promotion operator in the case when the irreducible representation is indexed by a rectangular partition. We prove a generalisation of this theorem in two directions: on the one hand we lift the restriction on the shape of the partition, and on the other hand we enlarge the result to the collection of all separable permutations.

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arXiv2019

On a conjecture of Pappas and Rapoport about the standard local model for $GL_d$

In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of $n \times n$ matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

math.AG↗
arXiv2019

On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products

Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights $λ$ and $μ$ for a Lie algebra $\mathfrak{g}$, which we will assume is simply-laced. In this paper, we relate the category $\mathcal{O}$ over truncated shifted Yangians to categorified tensor products: for a generic integral choice of parameters, category $\mathcal{O}$ is equivalent to a weight space in the categorification of a tensor product of fundamental representations defined by the third author using KLRW algebras. We also give a precise description of category $\mathcal{O}$ for arbitrary parameters using a new algebra which we call the parity KLRW algebra. In particular, we confirm the conjecture of the authors that the highest weights of category $\mathcal{O}$ are in canonical bijection with a product monomial crystal depending on the choice of parameters. This work also has interesting applications to classical representation theory. In particular, it allows us to give a classification of simple Gelfand-Tsetlin modules of $U(\mathfrak{gl}_n)$ and its associated W-algebras.

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arXiv2019

Highest weights for truncated shifted Yangians and product monomial crystals

Truncated shifted Yangians are a family of algebras which are natural quantizations of slices in the affine Grassmannian. We study the highest weight representations of these algebras. In particular, we conjecture that the possible highest weights for these algebras are described by product monomial crystals, certain natural subcrystals of Nakajima's monomials. We prove this conjecture in type A. We also place our results in the context of symplectic duality and prove a conjecture of Hikita in this situation.

math.RT↗
arXiv2019

A quantum Mirković-Vybornov isomorphism

We present a quantization of an isomorphism of Mirković and Vybornov which relates the intersection of a Slodowy slice and a nilpotent orbit closure in $\mathfrak{gl}_N$ , to a slice between spherical Schubert varieties in the affine Grassmannian of $PGL_n$ (with weights encoded by the Jordan types of the nilpotent orbits). A quantization of the former variety is provided by a parabolic W-algebra and of the latter by a truncated shifted Yangian. Building on earlier work of Brundan and Kleshchev, we define an explicit isomorphism between these non-commutative algebras, and show that its classical limit is a variation of the original isomorphism of Mirković and Vybornov. As a corollary, we deduce that the W-algebra is free as a left (or right) module over its Gelfand-Tsetlin subalgebra, as conjectured by Futorny, Molev, and Ovsienko.

math.RT↗
arXiv2018

The equations defining affine Grassmannians in type A and a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman

The affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part of the ideal defining this embedding in terms of certain elements of the dual of Fock space called "shuffles", and they conjecture that these elements together with the Plücker relations suffice to cut out the affine Grassmannian. We give a proof of this conjecture in two steps: first we reinterpret the shuffles equations in terms of Frobenius twists of symmetric functions. Using this, we reduce to a finite dimensional-problem, which we solve. For the second step we introduce a finite-dimensional analogue of the affine Grassmannians of $SL_n$, which we conjecture to be precisely the reduced subscheme of a finite-dimensional Grassmannian consisting of subspaces invariant under a nilpotent operator.

math.AG↗
arXiv2017

An equivalence between truncations of categorified quantum groups and Heisenberg categories

We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $\mathfrak{sl}_\infty$. We show that $\mathscr{A}$ is equivalent to a truncation of the Khovanov--Lauda categorified quantum group $\mathscr{U}$ of type $A_\infty$, and also to a truncation of Khovanov's Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of the principal realization of the basic representation of $\mathfrak{sl}_\infty$. As a result of the categorical equivalences described above, certain actions of $\mathscr{H}$ induce actions of $\mathscr{U}$, and vice versa. In particular, we obtain an explicit action of $\mathscr{U}$ on representations of symmetric groups. We also explicitly compute the Grothendieck group of the truncation of $\mathscr{H}$. The 2-category $\mathscr{A}$ can be viewed as a graphical calculus describing the functors of $i$-induction and $i$-restriction for symmetric groups, together with the natural transformations between their compositions. The resulting computational tool is used to give simple diagrammatic proofs of (apparently new) representation theoretic identities.

math.RT↗
arXiv2017

Quantum Polynomial Functors

We construct a category of quantum polynomial functors which deforms Friedlander and Suslin's category of strict polynomial functors. The main aim of this paper is to develop from first principles the basic structural properties of this category (duality, projective generators, braiding etc.) in analogy with classical strict polynomial functors. We then apply the work of Hashimoto and Hayashi in this context to construct quantum Schur/Weyl functors, and use this to provide new and easy derivations of quantum $(GL_m,GL_n)$ duality, along with other results in quantum invariant theory.

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