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Hebing Rui

Publications and source records attributed to Hebing Rui.

At least 19 recordsLinked to original sources
arXiv2026

Affine and cyclotomic Brauer categorification via $\imath$-Kac--Moody $2$-categories: the half-integral type $\operatorname{AIII}$ case

Cyclotomic Brauer or cyclotomic Nazarov--Wenzl algebras arise in higher Schur--Weyl dualities involving parabolic categories $\mathcal O$ for Lie algebras of types $B,C$ and $D$. Their connection with Kac--Moody-type categorification is substantially less developed than the corresponding type $A$ theory for cyclotomic Hecke algebras. We construct a categorical bridge between affine Brauer-type representation theory and the half-integral quasi-split type $\operatorname{AIII}$ $\imath$-Kac--Moody $2$-category of Bao--Shan--Wang--Webster. More precisely, an action of the affine Brauer category on a locally Schurian category, with dot spectrum exactly $\frac{1}{2}+\mathbb Z$, determines a generalized nilpotent $2$-representation of the even component $\mathfrak U^{\imath}_{+}$. Conversely, every nilpotent $2$-subrepresentation of an ambient locally Schurian $2$-representation of $\mathfrak U^{\imath}_{+}$ carries a compatible affine Brauer action whose dot spectrum is contained in $\frac{1}{2}+\mathbb Z$. Applying these constructions to cyclotomic quotients, we prove that the locally unital algebra attached to a $\mathbf u$-admissible cyclotomic Brauer category is isomorphic to the locally unital algebra attached to the corresponding cyclotomic quotient of the principal $2$-representation of $\mathfrak U^{\imath}_{+}$. Consequently, the associated cyclotomic Brauer (or cyclotomic Nazarov--Wenzl) algebras acquire natural $\mathbb Z$-gradings. To our knowledge, this is the first such categorical realization of $\mathbf u$-admissible half-integral cyclotomic Brauer algebras by means of an $\imath$-Kac--Moody $2$-category. It provides the categorical and graded framework toward a Brauer-type extension of the Brundan--Kleshchev--Ariki theory in which coideal algebras and $\imath$-canonical bases are expected to replace ordinary quantum groups and canonical bases.

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arXiv2025

Decomposition numbers of the cyclotomic Brauer algebra over the complex field, II

Following Nazarov's suggestion, the cyclotomic Nazarov-Wenzl algebra is referred to as the cyclotomic Brauer algebra. This paper focuses on computing the decomposition numbers of the cyclotomic Brauer algebra over $\mathbb{C}$ with arbitrary parameters. We show that these decomposition numbers can be expressed in terms of the parabolic Kazhdan-Lusztig polynomials of type $D_n$, with a parabolic subgroup of type $A$, under Condition 1.2.

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arXiv2025

Decomposition numbers of cyclotomic Brauer algebras over the complex field, I

Following Nazarov's suggestion~\cite{Naz1}, we refer to the cyclotomic Nazarov-Wenzl algebra as the cyclotomic Brauer algebra. When the cyclotomic Brauer algebra is isomorphic to the endomorphism algebra of $M_{I_i, r}$-- the tensor product of a simple scalar-type parabolic Verma module with the natural module in the parabolic BGG category $\mathcal O$ of types $B_n$, $C_n$ and $D_n$, its decomposition numbers can theoretically be computed, based on general results from \cite{AST} and \cite[Corollary~5.10]{RS}. This paper aims to establish explicit connections between the parabolic Verma modules that appear as subquotients of $M_{I_i, r}$ and the right cell modules of the cyclotomic Brauer algebra under condition~\eqref{simple111}. It allows us to explicitly decompose $M_{I_i, r}$ into a direct sum of indecomposable tilting modules by identifying their highest weights and multiplicities. Our result demonstrates that the decomposition numbers of such a cyclotomic Brauer algebra can be explicitly computed using the parabolic Kazhdan-Lusztig polynomials of types $B_n$, $C_n$, and $D_n$ with suitable parabolic subgroups~\cite{So}. Finally, condition~\eqref{simple111} is well-supported by a result of Wei Xiao presented in Section~6.

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arXiv2023

Affine Brauer category and parabolic category $\mathcal O$ in types $B, C, D$

A strict monoidal category referred to as affine Brauer category $\mathcal{AB}$ is introduced over a commutative ring $\kappa$ containing multiplicative identity $1$ and invertible element $2$. We prove that morphism spaces in $\mathcal{AB}$ are free over $\kappa$. The cyclotomic (or level $k$) Brauer category $\mathcal{CB}^f(\omega)$ is a quotient category of $\mathcal{AB}$. We prove that any morphism space in $\mathcal{CB}^f(\omega)$ is free over $\kappa$ with maximal rank if and only if the $\mathbf u$-admissible condition holds in the sense of (1.30). Affine Nazarov-Wenzl algebras and cyclotomic Nazarov-Wenzl algebras will be realized as certain endomorphism algebras in $\mathcal{AB}$ and $\mathcal{CB}^f(\omega)$, respectively. We will establish higher Schur-Weyl duality between cyclotomic Nazarov-Wenzl algebras and parabolic BGG categories $\mathcal O$ associated to symplectic and orthogonal Lie algebras over the complex field $\mathbb C$. This enables us to use standard arguments in [1,26,27] to compute decomposition matrices of cyclotomic Nazarov-Wenzl algebras. The level two case was considered by Ehrig and Stroppel in [14].

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arXiv2023

Representations of Brauer category and categorification

We study representations of the locally unital and locally finite dimensional algebra $B$ associated to the Brauer category $\mathcal B(\delta_0)$ with defining parameter $\delta_0$ over an algebraically closed field $K$ with characteristic $p\neq 2$. The Grothendieck group $K_0(B\text{-mod}^\Delta)$ will be used to categorify the integrable highest weight $\mathfrak {sl}_{K}$-module $ V(\varpi_{\frac{\delta_0-1}{2}})$ with the fundamental weight $\varpi_{\frac{\delta_0-1}{2}}$ as its highest weight, where $B$-mod$^\Delta$ is a subcategory of $B$-lfdmod in which each object has a finite $\Delta$-flag, and $\mathfrak {sl}_{K}$ is either $\mathfrak{sl}_\infty$ or $\hat{\mathfrak{sl}}_p$ depending on whether $p=0$ or $2\nmid p$. As $\mathfrak g$-modules, $\mathbb C\otimes_{\mathbb Z} K_0(B\text{-mod}^\Delta)$ is isomorphic to $ V(\varpi_{\frac{\delta_0-1}{2}})$, where $\mathfrak g$ is a Lie subalgebra of $\mathfrak {sl}_{K}$ (see Definition~4.2). When $p=0$, standard $B$-modules and projective covers of simple $B$-modules correspond to monomial basis and so-called quasi-canonical basis of $V(\varpi_{\frac{\delta_0-1}{2}}) $, respectively.

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arXiv2023

Blocks of the Brauer category over the complex field

Let $\mathcal B(\delta)$ be the Brauer category over the complex field $\mathbb C$ with the parameter $\delta$. In non-semisimple case, $\delta$ is an integer, and each weight space of $(\frac{\delta}2-1)$th semi-infinite wedge space corresponds to either a single block or a union of two different blocks of $\mathcal B(\delta)$-lfdmod, the category of the locally finite-dimensional representations of $\mathcal B(\delta)$. Furthermore, each block contains an infinite number of irreducible representations of $\mathcal B(\delta)$, and all blocks of $\mathcal B(\delta)$-lfdmod can be obtained in this way

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arXiv2022

The periplectic $q$-Brauer category

We introduce the periplectic $q$-Brauer category over an integral domain of characteristic not $2$. This is a strict monoidal supercategory and can be considered as a $q$-analogue of the periplectic Brauer category. We prove that the periplectic $q$-Brauer category admits a split triangular decomposition in the sense of Brundan-Stroppel. When the ground ring is an algebraically closed field, the category of locally finite dimensional right modules for the periplectic $q$-Brauer category is an upper finite fully stratified category in the sense of Brundan and Stroppel. We prove that periplectic $q$-Brauer algebras defined in [1] are isomorphic to endomorphism algebras in the periplectic $q$-Brauer category. Furthermore, a periplectic $q$-Brauer algebra is a standardly based algebra in the sense of Du and Rui. We construct Jucys-Murphy basis for any standard module of the periplectic $q$-Brauer algebra with respect to a family of commutative elements called Jucys-Murphy elements. Via them, we classify blocks for both periplectic $q$-Brauer category and periplectic $q$-Brauer algebras in generic case. Our result shows that both periplectic $q$-Brauer category and periplectic $q$-Brauer algebras are always not semisimple over any algebraically closed field.

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arXiv2021

Representations of cyclotomic oriented Brauer categories

Let $A$ be the locally unital algebra associated to a cyclotomic oriented Brauer category over an arbitrary algebraically closed field $\Bbbk$ of characteristic $p\ge 0$. The category of locally finite dimensional representations of $A $ is used to give the tensor product categorification (in the general sense of Losev and Webster) for an integrable lowest weight with an integrable highest weight representation of the same level for the Lie algebra $\mathfrak g$, where $\mathfrak g$ is a direct sum of copies of $\mathfrak {sl}_\infty$ (resp., $ \hat{\mathfrak {sl}}_p$ ) if $p=0$ (resp., $p>0$). Such a result was expected in [3] when $\Bbbk=\mathbb C$ and proved previously by Brundan in [2] when the level is $1$.

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arXiv2020

Representations of weakly triangular categories

A new class of locally unital and locally finite dimensional algebras $A$ over an arbitrary algebraically closed field is discovered. Each of them admits an upper finite weakly triangular decomposition, a generalization of an upper finite triangular decomposition. Any locally unital algebra which admits an upper finite Cartan decomposition is Morita equivalent to some special locally unital algebra $A$ which admits an upper finite weakly triangular decomposition. It is established that the category $A$-lfdmod of locally finite dimensional left $A$-modules is an upper finite fully stratified category in the sense of Brundan-Stroppel. Moreover, $A$ is semisimple if and only if its centralizer subalgebras associated to certain idempotent elements are semisimple. Furthermore, certain endofunctors are defined and give categorical actions of some Lie algebras on the subcategory of $A$-lfdmod consisting of all objects which have a finite standard filtration. In the case $A$ is the locally unital algebra associated to one of cyclotomic oriented Brauer categories, cyclotomic Brauer categories and cyclotomic Kauffman categories, $A$ admits an upper finite weakly triangular decomposition. This leads to categorifications of representations of the classical limits of coideal algebras, which come from all integrable highest weight modules of $\mathfrak {sl}_\infty$ or $\hat {\mathfrak{sl}}_e$. Finally, we study representations of $A$ associated to either cyclotomic Brauer categories or cyclotomic Kauffman categories in details, including explicit criteria on the semisimplicity of $A$ over an arbitrary field, and on $A$-lfdmod being upper finite highest weight category in the sense of Brundan-Stroppel, and on Morita equivalence between $A$ and direct sum of infinitely many (degenerate) cyclotomic Hecke algebras.

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arXiv2020

A basis theorem for the affine Kauffmann category and its cyclotomic quotients

The affine Kauffmann category is a strict monoidal category and can be considered as a $q$-analogue of the affine Brauer category in (Rui et al. in Math. Zeit. 293, 503-550, 2019). In this paper, we prove a basis theorem for the morphism spaces in the affine Kauffmann category. The cyclotomic Kauffmann category is a quotient category of the affine Kauffmann category. We also prove that any morphism space in this category is free over an integral domain $\mathbb K$ with maximal rank if and only if the $\mathbf u$-admissible condition holds in the sense of Definition 1.13.

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arXiv2018

A proof of Comes-Kujawa's conjecture

Let $\kappa$ be a commutative ring containing $2^{-1}$. In this paper, we prove the Comes-Kujawa's conjecture on a $\kappa$-basis of cyclotomic oriented Brauer-Clifford supercategory. As a by-product, we prove that the cyclotomic walled Brauer-Clifford superalgebra defined by Comes and Kujawa and ours are isomorphic if $\kappa$ is an algebraically closed field with characteristic not two.

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arXiv2017

Affine walled Brauer-Clifford superalgebras

In this paper, a notion of affine walled Brauer-Clifford superalgebras $BC_{r, t}^{\rm aff} $ is introduced over an arbitrary integral domain $R$ containing $2^{-1}$. These superalgebras can be considered as affinization of walled Brauer superalgebras in \cite{JK}. By constructing infinite many homomorphisms from $BC_{r, t}^{\rm aff}$ to a class of level two walled Brauer-Clifford superagebras over $\mathbb C$, we prove that $BC_{r, t}^{\rm aff} $ is free over $R$ with infinite rank. We explain that any finite dimensional irreducible $BC_{r, t}^{\rm aff} $-module over an algebraically closed field $F$ of characteristic not $2$ factors through a cyclotomic quotient of $BC_{r, t}^{\rm aff} $, called a cyclotomic (or level $k$) walled Brauer-Clifford superalgebra $ BC_{k, r, t}$. Using a previous method on cyclotomic walled Brauer algebras in \cite{RSu1}, we prove that $BC_{k, r, t}$ is free over $R$ with super rank $(k^{r+t}2^{r+t-1} (r+t)!, k^{r+t}2^{r+t-1} (r+t)!)$ if and only if it is admissible in the sense of Definition~6.4. Finally, we prove that the degenerate affine (resp., cyclotomic) walled Brauer-Clifford superalgebras defined by Comes-Kujawa in \cite{CK} are isomorphic to our affine (resp., cyclotomic) walled Brauer-Clifford superalgebras.

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arXiv2015

Mixed Schur-Weyl duality between general linear Lie algebras and cyclotomic walled Brauer algebras

Motivated by Brundan-Kleshchev's work on higher Schur-Weyl duality, we establish mixed Schur-Weyl duality between general linear Lie algebras and cyclotomic walled Brauer algebras in an arbitrary level. Using weakly cellular bases of cyclotomic walled Brauer algebras, we classify highest weight vectors of certain mixed tensor modules of general linear Lie algebras. This leads to an efficient way to compute decomposition matrices of cyclotomic walled Brauer algebras arising from mixed Schur-Weyl duality, which generalizes early results on level two walled Brauer algebras.

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arXiv2014

Decomposition numbers of quantized walled Brauer algebras

In this paper, we establish explicit relationship between decomposition numbers of quantized walled Brauer algebras and those for either Hecke algebras associated to certain symmetric groups or (rational) $q$-Schur algebras over a field $κ$. This enables us to use Ariki's result \cite{Ar} and Varagnolo-Vasserot's result \cite{VV} to compute such decomposition numbers via inverse Kazhdan-Lusztig polynomials associated with affine Weyl groups of type $A$ if the ground field is $\mathbb C$.

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