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Chun-Ju Lai

Publications and source records attributed to Chun-Ju Lai.

18 recordsLinked to original sources
arXiv2026

On Cellularity of Hecke Algebras for Wreath Products

The (generalized) Hu algebra is a nontrivial quantization of the wreath product $Σ_m \wr Σ_d$ between symmetric groups, whose representation theory controls the Hecke algebra of the complex reflection group $G(d,d,md)$. In this paper, we construct a unified basis for this algebra and establish its cellular algebra structure in the case $d = 2$. As an application, our construction provides an elementary realization of the simple modules for the Hecke algebra of type $D_{2m}$ that are parameterized by bipartitions of size $(m,m)$.

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arXiv2026

Schurification of polynomial quantum wreath products

We study the Schur algebra counterpart of a vast class of quantum wreath products. This is achieved by developing a theory of twisted convolution algebras, inspired by geometric intuition. In parallel, we provide an algebraic Schurification via a Kashiwara-Miwa-Stern-type action on a tensor space. We give a uniform proof of Schur duality, and construct explicit bases of the new Schur algebras. This provides new results for, among other examples, Vignéras' pro-$p$ Iwahori Hecke algebras of type $A$, degenerate affine Hecke algebras, Kleshchev-Muth's affine zigzag algebras, and Rosso-Savage's affine Frobenius Hecke algebras.

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arXiv2026

Quantum wreath products and Schur--Weyl duality II

In the first part of this series, the authors introduced the quantum wreath product, providing a unified framework that encompasses numerous results previously addressed only through case-by-case analysis. This paper shifts focus to the fundamental construction of modules over these products, termed wreath modules. Our approach utilizes parabolic induction on tensor products combined with a sophisticated labeling scheme based on multipartitions. While the underlying constructions are technically involved, they offer a transparent realization of several prominent module families. Specifically, these wreath modules recover and unify: Simple modules over the Ariki-Koike algebra; Specht and simple modules over the Hu algebra; (anti)spherical modules and Kashiwara-Miwa-Stern modules over the affine Hecke algebra and its pro-p Iwahori variants. Finally, we demonstrate that these wreath modules for the Hu algebra serve as a critical component in solving the Ginzburg-Guay-Opdam-Rouquier problem. This solution enables a concrete realization of Category O for the rational Cherednik algebra in Type D.

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arXiv2026

On the Springer correspondence for wreath products

We establish a Bruhat decomposition indexed by the wreath product $Σ_m\wr Σ_d$ between two symmetric groups -- note that $Σ_m\wr Σ_d$ is not a Coxeter group in general. We show that such a decomposition affords a geometric variant in terms of the Bialynicki-Birula decomposition for varieties with $\mathbb{C}^*$-actions. Next, we construct a Steinberg variety whose top Borel-Moore homology realizes the group algebra $\mathbb{Q}[Σ_m\wr Σ_d]$ as a proper subalgebra. Such a geometric realization leads to a Springer-type correspondence which identifies the irreducible representations of $Σ_m\wr Σ_d$ with isotypic components of certain unconventional Springer fibers using type A geometry. In other words, we obtain a geometric counterpart of the (algebraic) Clifford theory, for the first time. Consequently, we obtain a new Springer correspondence of Weyl groups of type B/C/D using essentially type A geometry.

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arXiv2026

Quantum wreath products and $p$-adic general linear group

We study the pro-$p$ Iwahori-Hecke algebra and its Gelfand-Graev modules for the $p$-adic general linear group and its metaplectic covers. We develop the theory of quantum wreath products of skew polynomial type and use it to provide transparent descriptions of these Hecke algebras and their modules that were previously inaccessible through standard $p$-adic methods. We introduce the notion of (anti)spherical and Kashiwara-Miwa-Stern modules for these quantum wreath products for the first time and interpret the $p$-adic Gelfand-Graev modules in terms of these new modules. As an application, we study the structure theory for the corresponding $p$-adic pro-$p$ Schur algebras and obtain an explicit basis and multiplication rules. Moreover, we give algebraic (re)proofs of several results of $p$-adic interest including the existence of PBW basis for the pro-$p$ metaplectic and Iwahori-Hecke algebras, identification of the Iwahori-Schur algebra with the quantum affine Schur algebra, and failure of the local Shimura correspondence at the pro-$p$ level.

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arXiv2025

Category $\mathcal{O}$ for Lie superalgebras

The authors define a Category $\mathcal{O}$ for any quasi-reductive Lie superalgebra $\mathfrak{g}$ with respect to a triangular decomposition. This much needed approach unifies many important constructions in the existing literature in a rigorous fashion. Our Category $\mathcal{O}$ encompasses all highest weight categories for Lie (super)algebras as well as specific examples which may not be highest weight categories. When the decomposition arises from a principal parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{g}$, the Category $\mathcal{O}$ exhibits rich homological properties. For one, the authors show that in contrast to the case of a semisimple Lie algebra, the Category $\mathcal{O}$ is standardly stratified. Furthermore, the categorical cohomology of $\mathcal{O}$ is a finitely generated ring. This provides a first step towards developing a support variety theory for Category $\mathcal{O}$. It is shown that the complexity of modules in Category $\mathcal{O}$ is finite with an explicit upper bound given by the dimension of the subspace of the odd degree elements in $\mathfrak{g}$. This upgrades results known for $\mathfrak{gl}(m|n)$ to the more general setting. Our arguments are based on foundational connections between the categorical cohomology and the relative Lie superalgebra cohomology as well as the interplay between Category $\mathcal{O}$ for $\mathfrak{g}$ and the Category $\mathcal{O}$ for its corresponding Lie algebra $\mathfrak{g}_{\bar 0}$.

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arXiv2025

Infinite-Dimensional Towers and a Categorification of Differential Operators on $\mathbb{A}^1_{\mathbb{Z}[v,v^{- 1}]}$

We introduce axioms for towers of infinite-dimensional algebras such that the corresponding Grothendieck groups of projective and finite-dimensional modules are Hopf dual to each other. This duality gives rise to an action of the Hesienberg double -- a generalization of the classical Hesienberg algebra -- on the Grothendieck group. We categorify this action and, as an application, construct a categorical realization of quantum differential operators on $\mathbb{A}^1_{\mathbb{Z}[v,v^{-1}]}$.

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arXiv2025

Affine flag varieties and quantum symmetric pairs

The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown to be a coideal subalgebra of the quantum group of finite type $A$. In this paper we study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type $C$. We show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine $\mathfrak{sl}$ and $\mathfrak{gl}$ types, respectively. In this way, we provide geometric realizations of eight quantum symmetric pairs of affine types. We construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine $\mathfrak{sl}$ type, we establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, we obtain a new and geometric construction of the idempotented quantum affine $\mathfrak{gl}$ and its canonical basis.

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arXiv2024

Quantum wreath products and Schur-Weyl duality I

In this paper the authors introduce a new notion called the quantum wreath product, which is the algebra $B \wr_Q \mathcal{H}(d)$ produced from a given algebra $B$, a positive integer $d$, and a choice $Q=(R,S,ρ,σ)$ of parameters. Important examples {that arise from our construction} include many variants of the Hecke algebras, such as the Ariki-Koike algebras, the affine Hecke algebras and their degenerate version, Wan-Wang's wreath Hecke algebras, Rosso-Savage's (affine) Frobenius Hecke algebras, Kleshchev-Muth's affine zigzag algebras, and the Hu algebra that quantizes the wreath product $Σ_m \wr Σ_2$ between symmetric groups. In the first part of the paper, the authors develop a structure theory for the quantum wreath products. Necessary and sufficient conditions for these algebras to afford a basis of suitable size are obtained. Furthermore, a Schur-Weyl duality is established via a splitting lemma and mild assumptions on the base algebra $B$. Our uniform approach encompasses many known results which were proved in a case by case manner. The second part of the paper involves the problem of constructing natural subalgebras of Hecke algebras that arise from wreath products. Moreover, a bar-invariant basis of the Hu algebra via an explicit formula for its extra generator is also described.

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arXiv2022

Irreducible components of two-row Springer fibers for all classical types

We give an explicit description of the irreducible components of two-row Springer fibers for all classical types using cup diagrams. Cup diagrams can be used to label the irreducible components of two-row Springer fibers. Given a cup diagram, we explicitly write down all flags contained in the component associated to the cup diagram. This generalizes results by Stroppel--Webster and Fung to all classical types.

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arXiv2021

A study of irreducible components of Springer fibers using quiver varieties

It is a remarkable theorem by Maffei--Nakajima that the Slodowy variety, which is a subvariety of the resolution of the nilpotent cone, can be realized as a Nakajima quiver variety of type A. However, the isomorphism is rather implicit as it takes to solve a system of equations in which the variables are linear maps. In this paper, we construct solutions to this system under certain assumptions. This establishes an explicit and efficient way to compute the image of a complete flag contained in the Slodowy variety under the Maffei--Nakajima isomorphism and describe these flags in terms of quiver representations. As Slodowy varieties contain Springer fibers naturally, we can use these results to provide an explicit description of the irreducible components of two-row Springer fibers in terms of a family of kernel relations via quiver representations.

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arXiv2020

Irreducible components of two-row Springer fibers and Nakajima quiver varieties

We give an explicit description of the irreducible components of two-row Springer fibers in type A as closed subvarieties in certain Nakajima quiver varieties in terms of quiver representations. By taking invariants under a variety automorphism, we obtain an explicit algebraic description of the irreducible components of two-row Springer fibers of classical type. As a consequence, we discover relations on isotropic flags that describe the irreducible components.

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arXiv2019

Affine Hecke algebras and quantum symmetric pairs

We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra ${\mathbf K}^{\mathfrak c}_n$. We show that ${\mathbf K}^{\mathfrak c}_n$ is a coideal subalgebra of quantum affine algebra ${\bf U}(\hat{\mathfrak{gl}}_n)$, and $\big({\mathbf U}(\hat{ \mathfrak{gl}}_n), {\mathbf K}^{\mathfrak c}_n)$ forms a quantum symmetric pair. The modified coideal subalgebra is shown to admit monomial and stably canonical bases. We also formulate several variants of the affine Schur algebra and the (modified) coideal subalgebra above, as well as their monomial and canonical bases. This work provides a new and algebraic approach which complements and sheds new light on our previous geometric approach on the subject. In the appendix by four of the authors, new length formulas for the Weyl groups of affine classical types are obtained in a symmetrized fashion.

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arXiv2019

On $q$-Schur algebras corresponding to Hecke algebras of type B

In this paper the authors investigate the $q$-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type A. The authors present a coordinate algebra type construction that allows us to realize these $q$-Schur algebras as the duals of the $d$th graded components of certain graded coalgebras. Under suitable conditions an isomorphism theorem is proved that demonstrates that the representation theory reduces to the $q$-Schur algebra of type A. This enables the authors to address the questions of cellularity, quasi-hereditariness and representation type of these algebras. Later it is shown that these algebras realize the $1$-faithful quasi hereditary covers of the Hecke algebras of type B. As a further consequence, the authors demonstrate that these algebras are Morita equivalent to Rouquier's finite-dimensional algebras that arise from the category ${\mathcal O}$ for rational Cherednik algebras for the Weyl group of type B. In particular, we have introduced a Schur-type functor that identifies the type B Knizhnik-Zamolodchikov functor.

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arXiv2019

Schur algebras and quantum symmetric pairs with unequal parameters

We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson-Lusztig-MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type A III/IV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated to any weight function. These bases are the counterparts of Lusztig's bar-invariant basis for Hecke algebras with unequal parameters. In the appendix we provide an algebraic version of a type D Beilinson-Lusztig-MacPherson construction which is first introduced by Fan-Li from a geometric viewpoint.

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arXiv2017

An elementary construction of monomial bases of modified quantum affine $\mathfrak{gl}_n$

In 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum $\mathfrak{gl}_n$ and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic construction of the canonical basis for modified quantum affine $\mathfrak{gl}_n$, which among other results used an earlier construction of monomial bases using Ringel-Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine $\mathfrak{gl}_n$.

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arXiv2016

On Weyl modules over affine Lie algebras in prime characteristic

We construct a family of homomorphisms between Weyl modules for affine Lie algebras in characteristic p, which supports our conjecture on the strong linkage principle in this context. We also exhibit a large class of reducible Weyl modules beyond level one, for p not necessarily small.

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