arXiv subjects

Yaolong Shen

Publications and source records attributed to Yaolong Shen.

15 recordsLinked to original sources

arXiv2026

Big categorification on towers of classical groups and wreath product groups

We develop a uniform framework for ``big'' categorification of representation categories of towers of finite classical groups and wreath product groups. We construct actions of symmetric products of Heisenberg categories, quantum in the finite classical group case and degenerate in the wreath product case. These actions lead to categorical actions of suitable symmetric products of Kac--Moody 2-categories, and hence to actions of large Lie algebras on Grothendieck groups. In characteristic zero, the resulting actions control the full graded centers of the group algebras through diagrammatic central elements, and the associated colored weight functions separate all irreducible ordinary characters. We also obtain modular block descriptions for wreath product groups through categorification.

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arXiv2026

A Serre-type presentation for $\imath$quantum supergroups of type A

We establish a Serre-type presentation for $\imath$quantum enveloping superalgebras arising from quantum supersymmetric pairs of type A. We first prove, in arbitrary type, that evaluating a quantum Serre polynomial on the coideal generators produces a correction of strictly smaller weight in the natural filtration using the projection technique. We then compute these remainder terms for all type A super Satake diagrams. In particular, we determine the new degree-four relations associated with isotropic odd simple roots of the relevant local Satake diagrams.

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arXiv2026

Quantized Coulomb Branches of Separated Cotangent Type and Orthosymplectic Quivers

We propose a definition of the quantized Coulomb branches of separated cotangent type, and prove that the corresponding classical construction recovers the non-cotangent Coulomb branch. We also obtain a formula for quasi-minuscule monopole operators in arbitrary cotangent type. Applying these results, we compute the monopole operators for orthosymplectic quivers and construct a homomorphism from the shifted twisted Yangian of split ADE type to the corresponding quantized Coulomb branch algebra.

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arXiv2026

On weight modules over truncated shifted iYangians

Truncated shifted iYangians are a family of algebras expected to quantize certain components of affine Grassmannian islices. We introduce orientifold KLRW (oKLRW) algebras associated with quivers with involution and establish their faithful polynomial representations and diagrammatic bases. We also define KLR iYangians using double reflective KLR diagrams and construct diagrammatic realizations of the iGKLO homomorphisms. For integral parameters, we introduce interval oKLRW algebras and prove an equivalence between integral weight modules over truncated shifted iYangians and nilpotent modules over the corresponding interval oKLRW algebras.

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arXiv2026

The disoriented skein and iquantum Brauer categories

We develop a diagrammatic approach to the representation theory of the quantum symmetric pairs corresponding to orthosymplectic Lie superalgebras inside general linear Lie superalgebras. Our approach is based on the disoriented skein category, which we define as a module category over the framed HOMFLYPT skein category. The disoriented skein category admits full incarnation functors to the categories of modules over the iquantum enveloping algebras corresponding to the quantum symmetric pairs, and it can be viewed as an interpolating category for these categories of modules. We define an equivalence of module categories between the disoriented skein category and the iquantum Brauer category (also known as the $q$-Brauer category), after endowing the latter with the structure of a module category over the framed HOMFLYPT skein category. The disoriented skein category has some advantages over the iquantum Brauer category, possessing duality structure and allowing the incarnation functors to be strict morphisms of module categories. Finally, we construct explicit bases for the morphism spaces of the disoriented skein and iquantum Brauer categories.

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arXiv2026

Relative braid group symmetries on quantum supersymmetric pairs of type sAIII

We introduce the relative Coxeter groupoid and construct intrinsic relative braid group symmetries for quantum supersymmetric pairs of type sAIII. These symmetries are constructed by establishing new intertwining properties of quasi $K$-matrices, which generalize the earlier non-super construction of Wang and the second author. We derive explicit formulas for these symmetries and prove that they satisfy the braid relations in the relative Coxeter groupoid.

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arXiv2026

Classifying submodules over monoidal categories

We study the classification of submodules of module categories over monoidal categories, extending ideas of Coulembier on the classification of tensor ideals in monoidal categories. We develop a framework that applies to module categories equipped with a twisted cylinder twist, a structure closely related to the twisted reflection equation and quantum symmetric pairs. Under mild assumptions, we establish an order-preserving bijection between submodules of a module category $\mathcal{M}$ and submodules of the path-algebra module $\mathcal{M}(1,-)$. We show that this correspondence is compatible with idempotent completion and analyze its behavior under decategorification to the split Grothendieck group, giving criteria for classification in terms of indecomposable objects. As an application, we study the disoriented skein category as a module category over the oriented skein category, describe its indecomposable objects, and obtain a complete classification of its submodules.

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arXiv2025

Quivers with Involutions and Shifted Twisted Yangians via Coulomb Branches

To a quiver with involution, we study the Coulomb branch of the 3d $\mathcal{N} = 4$ involution-fixed part of the quiver gauge theory. We show that there is an algebra homomorphism from the corresponding shifted twisted Yangian to the quantized Coulomb branch algebra. This gives a new instance of 3D mirror symmetries.

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arXiv2025

Affine and cyclotomic $q$-Schur categories via webs

We formulate two new $\mathbb Z[q,q^{-1}]$-linear diagrammatic monoidal categories, the affine $q$-web category and the affine $q$-Schur category, as well as their respective cyclotomic quotient categories. Diagrammatic integral bases for the Hom-spaces of all these categories are established. In addition, we establish the following isomorphisms, providing diagrammatic presentations of these $q$-Schur algebras for the first time: (i)~ the path algebras of the affine $q$-web category to R.~Green's affine $q$-Schur algebras, (ii)~ the path algebras of the affine $q$-Schur category to Maksimau-Stroppel's higher level affine $q$-Schur algebras, and most significantly, (iii)~ the path algebras of the cyclotomic $q$-Schur categories to Dipper-James-Mathas' cyclotomic $q$-Schur algebras.

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arXiv2025

Quantum supersymmetric pairs of basic types

We formulate and classify super Satake diagrams under a mild assumption, building on arbitrary Dynkin diagrams for finite-dimensional basic Lie superalgebras. We develop a theory of quantum supersymmetric pairs associated to the super Satake diagrams. We establish the quantum Iwasawa decomposition and construct quasi $K$-matrix associated with the quantum supersymmetric pairs. We also formulate a Schur duality between an $\imath$quantum supergroup (which is a new $q$-deformation of an ortho-symplectic Lie superalgebra) and the $q$-Brauer algebra.

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arXiv2024

Canonical bases of the oriented skein category

We develop a bar involution and canonical basis for every morphism space of the oriented skein category through a diagrammatic approach. In particular, our construction gives rise to Kazhdan-Lusztig type bases on quantized walled Brauer algebras.

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arXiv2023

Quantum supersymmetric pairs and $\imath$Schur duality of type AIII

We construct quantum supersymmetric pairs $({\bold U},{\bold U}^\imath)$ of type AIII and elucidate their fundamental properties. An $\imath$Schur duality between the $\imath$quantum supergroup ${\bold U}^\imath$ and the Hecke algebra of type B acting on a tensor space is established, providing a super generalization of the $\imath$Schur duality of type AIII. Additionally, we construct a (quasi) $K$-matrix for arbitrary parameters, which facilitates the realization of the Hecke algebra action on the tensor space.

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arXiv2023

Quasi-parabolic Kazhdan-Lusztig bases and reflection subgroups

Recently, Wang and the second author constructed a bar involution and canonical basis for a quasi-permutation module of the Hecke algebra associated to a type B Weyl group $W$, where the basis is parameterized by left cosets of a quasi-parabolic reflection subgroup in $W$. In this paper we provide an alternative approach to these constructions, and then generalize to Coxeter groups which contain a product of type B Weyl groups as a parabolic subgroup.

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arXiv2023

$\imath$Schur duality and Kazhdan-Lusztig basis expanded

Expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. We formulate an $\imath$Schur duality between an $\imath$quantum group of type AIII (allowing black nodes in its Satake diagram) and a Hecke algebra of type B acting on a tensor space, providing a common generalization of Jimbo-Schur duality and Bao-Wang's quasi-split $\imath$Schur duality. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the $\imath$canonical basis on the tensor space. An inversion formula for quasi-parabolic KL polynomials is established via the $\imath$Schur duality.

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arXiv2022

Canonical bases of $q$-Brauer algebras and $\imath$Schur dualities

Expanding the classical work of Kazhdan-Lusztig, we construct a bar involution and canonical bases on the $q$-Brauer algebra introduced by Wenzl. We define explicit actions of the $q$-Brauer algebra on the tensor spaces, and formulate $\imath$Schur dualities between the $q$-Brauer algebra and the $\imath$quantum groups of type AI and AII respectively.

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