arXiv subjects

Weideng Cui

Publications and source records attributed to Weideng Cui.

At least 19 recordsLinked to original sources

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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arXiv2023

Asymptotic Schur algebras and cellularity of q-Schur algebras

We prove that the q-Schur algebras of finite type introduced in [LW22] are cellular in the sense of Graham and Lehrer, which is a generalization of Geck's theorem on the cellularity of Hecke algebras of finite type. Moreover, we study special modules of the associated asymptotic Schur algebras and left cell representations of Schur algebras, which generalize Lusztig's work about special modules of asymptotic Hecke algebras and left cell representations of Weyl groups, respectively.

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arXiv2022

Cells in modified $\imath$quantum groups of type AIII and related Schur algebras

We provide a combinatorial characterization of two-sided cells in modified $\imath$quantum groups of type AIII. Our approach is to lift a corresponding description of two-sided cells in $\jmath$-Schur algebras associated to Iwahori--Hecke algebras of type $B$. We further give a combinatorial description of two-sided cells in other two kinds of Schur-type algebras, namely $\imath$-Schur algebras and $\tilde{\imath}$-Schur algebras.

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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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arXiv2020

Cells in affine q-Schur algebras

We develop algebraic and geometrical approaches toward canonical bases for affine q-Schur algebras of arbitrary type introduced in this paper. A duality between an affine q-Schur algebra and a corresponding affine Hecke algebra is established. We introduce an inner product on the affine q-Schur algebra, with respect to which the canonical basis is shown to be positive and almost orthonormal. We then formulate the cells and asymptotic forms for affine q-Schur algebras, and develop their basic properties analogous to the cells and asymptotic forms for affine Hecke algebras established by Lusztig. The results on cells and asymptotic algebras are also valid for q-Schur algebras of arbitrary finite type.

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arXiv2019

Modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras

We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type $A$ (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic $p$. Secondly, the modular branching rules for these algebras are obtained; moreover, the resulting modular branching graphs for cyclotomic Yokonuma-Hecke algebras are identified with crystal graphs of irreducible integrable representations of affine Lie algebras of type $A.$

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arXiv2017

Affine cellularity of affine Yokonuma-Hecke algebras

We establish an explicit algebra isomorphism between the affine Yokonuma-Hecke algebra $\widehat{Y}_{r,n}(q)$ and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type $A.$ As an application of this result, we show that $\widehat{Y}_{r,n}(q)$ is affine cellular in the sense of Koenig and Xi, and further prove that it has finite global dimension when the parameter $q$ is not a root of the Poincaré polynomial. As another application, we also recover the modular representation theory of $\widehat{Y}_{r,n}(q)$ previously obtained in [CW].

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arXiv2017

An isomorphism theorem for Lusztig algebras

In [Lu6] Lusztig defined a certain algebra $H,$ which is a direct sum of various algebras $H_{\mathfrak{o}}.$ We establish an explicit algebra isomorphism between each algebra $H_{\mathfrak{o}}$ and some matrix algebra with coefficients in the tensor product of an (affine or finite) Hecke algebra and a group algebra. We give an application.

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arXiv2017

On the presentation of Hecke-Hopf algebras for non-simply-laced type

Hecke-Hopf algebras were defined by A. Berenstein and D. Kazhdan. We give an explicit presentation of an Hecke-Hopf algebra when the parameter $m_{ij},$ associated to any two distinct vertices $i$ and $j$ in the presentation of a Coxeter group, equals $4,$ $5$ or $6$. As an application, we give a proof of a conjecture of Berenstein and Kazhdan when the Coxeter group is crystallographic and non-simply-laced. As another application, we show that another conjecture of Berenstein and Kazhdan holds when $m_{ij},$ associated to any two distinct vertices $i$ and $j,$ equals $4$ and that the conjecture does not hold when some $m_{ij}$ equals $6$ by giving a counterexample to it.

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arXiv2016

Representations of $0$-Yokonuma-Hecke algebras

We give two different approaches to classifying the simple modules of $0$-Yokonuma-Hecke algebras $Y_{r,n}(0)$ over an algebraically closed field of characteristic $p$ such that $p$ does not divide $r.$ Using the isomorphism between the $0$-Yokonuma-Hecke algebra and $0$-Ariki-Koike-Shoji algebra, we in fact give another way to obtain the simple modules of the latter, which was previously studied by Hivert, Novelli and Thibon (Adv. Math. $\bf{205}$ (2006) 504-548). In the appendix, we give the classification of simple modules of the nil Yokonuma-Hecke algebra.

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arXiv2016

An Iwahori-Matsumoto presentation of affine Yokonuma-Hecke algebras

We first present an Iwahori-Matsumoto presentation of affine Yokonuma-Hecke algebras $\widehat{Y}_{r,n}(q)$ to give a new proof of the fact, which was previously proved by Chlouveraki and Sécherre, that $\widehat{Y}_{r,n}(q)$ is a particular case of the pro-$p$-Iwahori-Hecke algebras defined by Vignéras, meanwhile, we give one application. Using the new presentation, we then give a third presentation of $\widehat{Y}_{r,n}(q),$ from which we immediately get an unexpected result, that is, the extended affine Hecke algebra of type $A$ is a subalgebra of the affine Yokonuma-Hecke algebra.

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arXiv2016

Calibrated representations of affine Yokonuma-Hecke algebras

Inspired by the work [Ra1], we directly give a complete classification of irreducible calibrated representations of affine Yokonuma-Hecke algebras $\widehat{Y}_{r,n}(q)$ over $\mathbb{C},$ which are indexed by $r$-tuples of placed skew shapes. We then develop several applications of this result. In the appendix, inspired by [Ru], we classify and construct irreducible completely splittable representations of degenerate affine Yokonuma-Hecke algebras $D_{r,n}$ and the wreath product $(\mathbb{Z}/r\mathbb{Z})\wr \mathfrak{S}_{n}$ over an algebraically closed field of characteristic $p> 0$ such that $p$ does not divide $r$.

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arXiv2016

Fusion procedure for cyclotomic BMW algebras

Inspired by the work [IMOg2], in this note, we prove that the pairwise orthogonal primitive idempotents of generic cyclotomic Birman-Murakami-Wenzl algebras can be constructed by consecutive evaluations of a certain rational function. In the appendix, we prove a similar result for generic cyclotomic Nazarov-Wenzl algebras.

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arXiv2016

A note on degenerate cyclotomic Yokonuma-Hecke algebras

In this note, we first prove that the degenerate cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}$ is cellular by constructing an explicit cellular basis. We then develop the fusion procedure for $Y_{r,n}^{d}$, that is, a complete set of pairwise orthogonal primitive idempotents for $Y_{r,n}^{d}$ is defined by consecutive evaluations of a certain rational function.

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arXiv2016

Cellularity of cyclotomic Yokonuma-Hecke algebras

We first give a direct proof of a basis theorem for the cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q).$ Our approach follows Kleshchev's, which does not use the representation theory of $Y_{r,n}^{d}(q),$ and so it is very different from that of [ChP2]. We also present two applications. Then we prove that the cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q)$ is cellular by constructing an explicit cellular basis, and show that the Jucys-Murphy elements for $Y_{r,n}^{d}(q)$ are JM-elements in the abstract sense. In the appendix, we shall develop the fusion procedure for $Y_{r,n}^{d}(q).$

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arXiv2016

Fusion procedure for Yokonuma-Hecke algebras

Inspired by the work [PA1], in this note, we prove that a complete set of pairwise orthogonal primitive idempotents of Yokonuma-Hecke algebras can be obtained by consecutive evaluations of a certain rational function.

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arXiv2016

An isomorphism theorem for degenerate cyclotomic Yokonuma-Hecke algebras and applications

Inspired by the work [PA], we establish an explicit algebra isomorphism between the degenerate cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q)$ and a direct sum of matrix algebras over tensor products of degenerate cyclotomic Hecke algebras of type $A$. We then develop several applications of this result, including a new proof of the modular representation theory of $Y_{r,n}^{d}(q)$, a semisimplicity criterion for it and cellularity of it. Moreover, we prove that $Y_{r,n}^{d}(q)$ is a symmetric algebra and determine the associated Schur elements by using the isomorphism theorem for it.

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arXiv2016

Yokonuma-Schur algebras

In this paper, we define the Yokonuma-Schur algebra $\text{YS}_{q}(r,n)$ as the endomorphism algebra of a permutation module for the Yokonuma-Hecke algebra $\text{Y}_{r,n}(q).$ We prove that $\text{YS}_{q}(r,n)$ is cellular by constructing an explicit cellular basis following the approach in [DJM], and we further show that it is a quasi-hereditary cover of $\text{Y}_{r,n}(q)$ in the sense of Rouquier following [HM2]. We also introduce the tilting modules for $\text{YS}_{q}(r,n).$ In the appendix, we define and study the cyclotomic Yokonuma-Schur algebra in a similar way.

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