From double affine Hecke algebras to quantized affine Schur algebras

Indecomposable $0$-Hecke modules for extended Schur functions

The extended Schur functions form a basis of quasisymmetric functions that contains the Schur functions. We provide a representation-theoretic interpretation of this basis by constructing $0$-Hecke modules whose quasisymmetric characteristics are the extended Schur functions. We further prove these modules are indecomposable.

A geometric Schur-Weyl duality for quotients of affine Hecke algebras

After establishing a geometric Schur-Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts have nice ideals coming from geometry, allowing duality for quotients. Some of the quotients of the positive affine Hecke algebra are then identified to some cyclotomic Hecke algebras and the geometric setting allows the construction of canonical bases.

环路Hecke代数的基与Schur-Weyl对偶

A basis and Schur-Weyl duality for the loop Hecke algebra

本文在温和参数条件下给出环路Hecke代数的新表示和一组基，用高阶线性重写理论与Dyck路径组合证明基的线性无关性并计算其基数，从而证明Damiani-Martin-Rowell猜想，并建立其与量子gl(1|1)负半部分的Schur-Weyl对偶解释。

多项式量子缠绕积的Schur化

Schurification of polynomial quantum wreath products

通过发展扭曲卷积代数理论，研究一类广泛量子缠绕积的Schur代数对应物，并基于张量空间上的Kashiwara-Miwa-Stern型作用给出代数Schur化，统一证明Schur对偶性并构造新Schur代数的显式基。

Multiparameter quantum Schur duality of type B

We establish a Schur type duality between a coideal subalgebra of the quantum group of type A and the Hecke algebra of type B with 2 parameters. We identify the $\imath$-canonical basis on the tensor product of the natural representation with Lusztig's canonical basis of the type B Hecke algebra with unequal parameters associated to a weight function.

The decomposition of 0-Hecke modules associated to quasisymmetric Schur functions

Recently Tewari and van Willigenburg constructed modules of the 0-Hecke algebra that are mapped to the quasisymmetric Schur functions by the quasisymmetric characteristic and decomposed them into a direct sum of certain submodules. We show that these submodules are indecomposable by determining their endomorphism rings.

$0$-Hecke modules for Young row-strict quasisymmetric Schur functions

We construct modules of the $0$-Hecke algebra whose images under the quasisymmetric characteristic map are the Young row-strict quasisymmetric Schur functions. This provides a representation-theoretic interpretation of this basis of quasisymmetric functions, answering a question of Mason and Niese (2015). Additionally, we classify when these modules are indecomposable.

B型Duplex Hecke代数

Duplex Hecke Algebras of type B

本文引入B型duplex Hecke代数，推广了A型相关代数，证明其在张量空间有自然表示，并建立B型Levi型q-Schur-Weyl对偶，揭示该代数与ı量子群的双中心化子性质。

Higher level affine Schur and Hecke algebras

We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.

q-Schur algebras and complex reflection groups, I

We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a q-Schur algebra (parameter not a half integer), providing thus character formulas for simple modules. We give some generalization to B_n(d). We prove an ``abstract'' translation principle. These results follow from the unicity of certain highest categories covering Hecke algebras. We also provide a semi-simplicity criterion for Hecke algebras of complex reflection groups.

The queer q-Schur superalgebra

As a natural generalization quantum Schur algebras associated with the Hecke algebra of the symmetric group, we introduce the quantum Schur superalgebra of type Q associated with the Hecke-Clifford superalgebra, which, by definition, is the endomorphism algebra of the induced module over the Hecke-Clifford superalgebra from certain permutation modules over the Hecke algebra of the symmetric group. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base change property for them. We will also identify the quantum Schur superalgebra of type…

Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebra of gl(2)

We obtain a presentation of quantum Schur algebras (over the field Q(v)) by generators and relations. This presentation is compatible with the usual presentation of the quantized universal enveloping algebra of the Lie algebra gl(2). We also locate the ``integral'' form of the quantum Schur algebra within the presented algebra and show it has a basis which is closely related to Lusztig's basis of the integral form of the quantized enveloping algebra.

Hecke代数的广义钩长与Schur元

Generalised hook lengths and Schur elements for Hecke algebras

本研究比较分拆钩长的两种推广，应用于Ariki-Koike代数模表示论，证明单模Schur元可被其广义核的Schur元整除，且A型Hecke代数下该值等于核与商的Schur元乘积。

Slim cyclotomic q-Schur algebras

We construct a new basis for a slim cyclotomic $q$-Schur algebra $\cysSr$ via symmetric polynomials in Jucys--Murphy operators of the cyclotomic Hecke algebra $\cysHr$. We show that this basis, labelled by matrices, is not the double coset basis when $\cysHr$ is the Hecke algebra of a Coxeter group, but coincides with the double coset basis for the corresponding group algebra, the Hecke algebra at $q=1$. As further applications, we then discuss the cyclotomic Schur--Weyl duality at the integral level. This also includes a category equivalence and a classification of simple objects.

Schurian-finiteness of blocks of type $A$ Hecke algebras II

For any algebra $A$ over an algebraically closed field $\mathbb{F}$, we say that an $A$-module $M$ is Schurian if $\mathrm{End}_A(M) \cong \mathbb{F}$. We say that $A$ is Schurian-finite if there are only finitely many isomorphism classes of Schurian $A$-modules, and Schurian-infinite otherwise. In this paper, we build on the work of Ariki and the second author to show that all blocks of type $A$ Hecke algebras of weight at least $2$ in quantum characteristic $e \geq 3$ are Schurian-infinite. This proves that if $e \geq 3$ then blocks of type $A$ Hecke algebras are Schurian-finite if and only …

Modules of the 0-Hecke algebra and quasisymmetric Schur functions

We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of Schur functions known as quasisymmetric Schur functions. Furthermore, we classify combinatorially which of these 0-Hecke modules are indecomposable. From here, we establish that the natural equivalence relation arising from our 0-Hecke action has equivalence classes that are isomorphic to subintervals of the weak Bruhat order on the symmetric group. Focussing on the equivalence classes containin…

Representations of Two-Parameter Quantum Groups and Schur-Weyl Duality

We determine the finite-dimensional simple modules for two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl_n and sl_n, and give a complete reducibility result. These quantum groups have a natural n-dimensional module V. We prove an analogue of Schur-Weyl duality in this setting: the centralizer algebra of the quantum group action on the k-fold tensor power of V is a quotient of a Hecke algebra for all n and is isomorphic to the Hecke algebra in case n\geq k.

Matrix units and Schur elements for the degenerate cyclotomic Hecke algebras

The paper uses the cellular basis of the (semi-simple) degenerate cyclotomic Hecke algebras to investigate these algebras exhaustively. As a consequence, we describe explicitly the "Young's seminormal form" and a orthogonal bases for Specht modules and determine explicitly the closed formula for the natural bilinear form on Specht modules and Schur elements for the degenerate cyclotomic Hekce algebras.

Degree and valuation of the Schur elements of cyclotomic Hecke algebras

Following the generalization of the notion of families of characters, defined by Lusztig for Weyl groups, to the case of complex reflection groups, thanks to the definition given by Rouquier, we show that the degree and the valuation of the Schur elements (functions A and a) remain constant on the "families" of the cyclotomic Hecke algebras of the exceptional complex reflection groups. The same result has already been obtained for the groups of the infinite series and for some special cases of exceptional groups.

BMW algebra, quantized coordinate algebra and type C Schur--Weyl duality

We prove an integral version of the Schur--Weyl duality between the specialized Birman--Murakami--Wenzl algebra $B_n(-q^{2m+1},q)$ and the quantum algebra associated to the symplectic Lie algebra sp_{2m}. In particular, we deduce that this Schur--Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang [Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra (1) 306 (2006), 138--174] in the symplectic case. As a byproduct, we show that, as $Z[q,q^{-1}]$-algebra, the quantized coordinate algebra defined by Kashiwara is…

A $q$-analogue of derivations on the tensor algebra and the $q$-Schur-Weyl duality

This paper presents a $q$-analogue of an extension of the tensor algebra given by the same author. This new algebra naturally contains the ordinary tensor algebra and the Iwahori-Hecke algebra type $A$ of infinite degree. Namely this algebra can be regarded as a natural mix of these two algebras. Moreover, we can consider natural "derivations" on this algebra. Using these derivations, we can easily prove the $q$-Schur-Weyl duality (the duality between the quantum enveloping algebra of the general linear Lie algebra and the Iwahori-Hecke algebra of type $A$).

Schur-Weyl reciprocity for the q-analogue of the alternating group

We establish Schur-Weyl reciprocity for the q-analogue of the alternating group. We analyze the sign q-permutation representation of the Hecke algebra on the tensor product of the super vector space V in detail, and examine its restriction to the q-analogue of the alternating group. In consequence, we find out that if the dimensions of even part and odd part of V are same, then the centralizer of the q-analogue of the alternating group is a Z_2-crossed product of the centralizer of the Hecke algebra and obtain Schur-Weyl reciprocity between them. Though the structure of the centralizer is more…

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 …

抛物Hecke代数的Kazhdan-Lusztig基及其在Schur-Weyl双曲面中的应用

Kazhdan-Lusztig bases of parabolic Hecke algebras and applications to Schur-Weyl duality

本文研究了抛物Hecke代数的Kazhdan-Lusztig基及其在Schur-Weyl双曲面中的应用，包括不可约表示的分类和核的构造。

Schur-Weyl duality for higher levels

We extend Schur-Weyl duality to an arbitrary level $l \geq 1$, the case $l=1$ recovering the classical duality between the symmetric and general linear groups. In general, the symmetric group is replaced by the degenerate cyclotomic Hecke algebra over $\C$ parametrized by a dominant weight of level $l$ for the root system of type $A_\infty$. As an application, we prove that the degenerate analogue of the quasi-hereditary cover of the cyclotomic Hecke algebra constructed by Dipper, James and Mathas is Morita equivalent to certain blocks of parabolic category $\mathcal{O}$ for the general linear…

类型B Hecke代数块的Schurian有限性

Schurian-finiteness of blocks of type B Hecke algebras

本文研究了类型B Hecke代数块的Schurian有限性，确定了非积分情况下的所有块和积分情况下的大部分块的有限性，剩余情况通过分解数和表示论方法进行分析。

分圆Hecke代数上的迹型与分圆Schur代数的余中心

Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras

本文在A型分圆Hecke代数上定义了统一的迹型，并构造了双基及分圆Schur代数余中心的显式基，证明其维数与基域、Hecke参数及分圆参数无关。

AIII型量子超对称对与$\imath$Schur对偶

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

本文构造了AIII型量子超对称对，建立了$\imath$量子超群与B型Hecke代数在张量空间上的$\imath$Schur对偶，并构造任意参数下的（拟）$K$-矩阵以实现Hecke代数作用。

A geometric realisation of 0-Schur and 0-Hecke algebras

We define a new product on orbits of pairs of flags in a vector space, using open orbits in certain varieties of pairs of flags. This new product defines an associative $\mathbb{Z}$-algebra, denoted by $G(n,r)$. We show that $G(n,r)$ is a geometric realisation of the 0-Schur algebra $S_0(n, r)$ over $\mathbb{Z}$, which is the $q$-Schur algebra $S_q(n,r)$ at q=0. We view a pair of flags as a pair of projective resolutions for a quiver of type $\mathbb{A}$ with linear orientation, and study $q$-Schur algebras from this point of view. This allows us to understand the relation between $q$-Schur al…

On the decomposition numbers of the Hecke algebra of type $D_n$ when $n$ is even

Let $n\geq 4$ be an even integer. Let $K$ be a field with $\cha K\neq 2$ and $q$ an invertible element in $K$ such that $\prod_{i=1}^{n-1}(1+q^i)\neq 0$. In this paper, we study the decomposition numbers over $K$ of the Iwahori--Hecke algebra $\HH_q(D_n)$ of type $D_n$. We obtain some equalities which relate its decomposition numbers with certain Schur elements and the decomposition numbers of various Iwahori--Hecke algebras of type $A$ with the same parameter $q$. When $\cha K=0$, this completely determine all of its decomposition numbers. The main tools we used are the Morita equivalence the…

偏微分Hecke代数、Schur-Weyl对偶性与Frobenius特征公式

Mirabolic Hecke algebras, Schur-Weyl duality and Frobenius character formulas

本文研究了偏微分Hecke代数的新的呈现方式，建立了其与量子群之间的Schur-Weyl对偶性，并推导出Frobenius特征公式及递归的Murnaghan-Nakayama规则。

The rectangular representation of the double affine Hecke algebra via elliptic Schur-Weyl duality

Given a module $M$ for the algebra $\mathcal{D}_{\mathtt{q}}(G)$ of quantum differential operators on $G$, and a positive integer $n$, we may equip the space $F_n^G(M)$ of invariant tensors in $V^{\otimes n}\otimes M$, with an action of the double affine Hecke algebra of type $A_{n-1}$. Here $G= SL_N$ or $GL_N$, and $V$ is the $N$-dimensional defining representation of $G$. In this paper we take $M$ to be the basic $\mathcal{D}_{\mathtt{q}}(G)$-module, i.e. the quantized coordinate algebra $M= \mathcal{O}_{\mathtt{q}}(G)$. We describe a weight basis for $F_n^G(\mathcal{O}_{\mathtt{q}}(G))$ com…

On enhanced reductive groups (I): Parabolic Schur algebras and the dualities related to degenerate double Hecke algebras

An enhanced algebraic group $\uG$ of $G=\GL(V)$ over $\bbc$ is a product variety $\GL(V)\times V$, endowed with an enhanced cross product. Associated with a natural tensor representation of $\uG$, there are naturally Levi and parabolic Schur algebras $\mathcal{L}$ and $\mathcal{P}$ respectively. We precisely investigate their structures, and study the dualities on the enhanced tensor representations for variant groups and algebras. In this course, an algebraic model of so-called degenerate double Hecke algebras (DDHA) is produced, and becomes a powerful implement. The connection between $\math…

Presenting Schur Algebras

Motivated by work of R.M. Green, we obtain a presentation of Schur algebras (both the classical and quantized versions) in terms of generators and relations. The presentation is compatible with the usual presentation of the (quantized or classical) enveloping algebra of gl(n). As a result, we obtain a new ``integral'' basis for Schur algebras which is a subset of Kostant's basis of the integral form of the enveloping algebra (or its q-analogue). Projection onto an appropriate component gives a new "integral" basis and a presentation for the Hecke algebra, compatible with the basis and presenta…

$\imath$Schur 对偶与 Kazhdan-Lusztig 基的展开

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

本文在 B 型 Hecke 代数的拟置换模上构造拟抛物 KL 基，建立 AIII 型 $\imath$量子群与 B 型 Hecke 代数的 $\imath$Schur 对偶，并证明该基与张量空间上的 $\imath$典范基一致，进而得到拟抛物 KL 多项式的反演公式。

Schur--Weyl 对偶的 Framization 与 Yokonuma--Hecke 型代数

Framization of Schur--Weyl duality and Yokonuma--Hecke type algebras

本文通过 Schur--Weyl 对偶思想研究代数的 framization，提供了一个使 Yokonuma--Hecke 等代数自然出现的一般性框架，并利用框架辫群构造了新的代数版本。

Quantum Schur duality of affine type C with three parameters

We establish a three-parameter Schur duality between the affine Hecke algebra of type C and a coideal subalgebra of quantum affine $\mathfrak{sl}_n$. At the equal parameter specializations, we obtain Schur dualities of types BCD.

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…

关于互反特征与量子仿射Schur-Weyl对偶

On reciprocal characters and the quantum affine Schur-Weyl duality

本文通过仿射Hecke代数的Langlands/Zelevinsky参数化识别了Frenkel-Reshetikhin q-特征的主导部分，并引入互反特征，在量子仿射Schur-Weyl对偶下证明其与量子仿射代数有限维模的主导q-特征匹配，实现了Nakajima关于主导单项式在Lusztig框架中作用的A型实现。

Quantized Algebras of Functions on Affine Hecke Algebras

The so called quantized algebras of functions on affine Hecke algebras of type A and the corresponding q-Schur algebras are defined and their irreducible unitarizable representations are classified.

mirabolic Hecke代数的量子超Schur–Weyl对偶与特征标公式

Quantum super Schur--Weyl duality and character formulas for mirabolic Hecke algebras

该研究建立了mirabolic Hecke代数的量子超Schur–Weyl对偶，推导了其不可约特征标的相关公式，还构造了超mirabolic RSK双射，并建立了其与q-rook幺半群代数的参数反转同构以转移相关结果。

Auslander-Reiten quiver of type A and generalized quantum affine Schur-Weyl duality

The quiver Hecke algebra $R$ can be also understood as a generalization of the affine Hecke algebra of type $A$ in the context of the quantum affine Schur-Weyl duality by the results of Kang, Kashiwara and Kim. On the other hand, it is well-known that the Auslander-Reiten(AR) quivers $Γ_Q$ of finite simply-laced types have a deep relation with the positive roots systems of the corresponding types. In this paper, we present explicit combinatorial descriptions for the AR-quivers $Γ_Q$ of finite type $A$. Using the combinatorial descriptions, we can investigate relations between finite dimensiona…

量子虚Schur-Weyl对偶

Quantum imaginary Schur-Weyl duality

本文建立了任意参数下无扭仿射A型箭袋Hecke代数与对称群Iwahori-Hecke代数之间的对偶，推广了Kleshchev和Muth的虚Schur-Weyl对偶，并用于计算简单模和标准模的特征。

量子圈积与Schur-Weyl对偶 I

Quantum wreath products and Schur-Weyl duality I

本文提出量子圈积的新概念，建立其结构理论与Schur-Weyl对偶，统一涵盖多类Hecke代数变体的已有结果，并构造Hecke代数的自然子代数、给出Hu代数的bar不变基。

Fusion Procedure for Cyclotomic Hecke Algebras

A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element.

Multiplication formulas and isomorphism theorem of $\imath$Schur superalgebras

We introduce the notion of $\imath$Schur superalgebra, which can be regarded as a type B/C counterpart of the $q$-Schur superalgebra (of type A) formulated as centralizer algebras of certain signed $q$-permutation modules over Hecke algebras. Some multiplication formulas for $\imath$Schur superalgebra are obtained to construct their canonical bases. Furthermore, we established an isomorphism theorem between the $\imath$Scuhr superalgebras and the $q$-Schur superalgebras of type A, which helps us derive a semisimplicity criteria of the $\imath$Schur superalgebras.

类型A的仿射Hecke和Schur代数无q平方根

Affine Hecke and Schur algebras of type A without a square root of q

本文提出了一种无需q平方根的仿射细胞结构，用于类型A的仿射Hecke和Schur代数，以支持p-adic域上GL_n(F)的表示理论研究。

$t$-量子化 Cartan 矩阵与 quiver Hecke 代数上 cuspidal 模的 R-矩阵

$t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras

本文证明任意有限型的 $t$-量子化 Cartan 矩阵可决定 quiver Hecke 代数 cuspidal 模间 R-矩阵的不变量，并利用 Dynkin quiver 的组合 AR-quiver 作为证明关键。

Diagram supermodules for $0$-Hecke-Clifford algebras

We introduce a general method for constructing modules for $0$-Hecke algebras and supermodules for $0$-Hecke-Clifford algebras from diagrams of boxes in the plane, and give formulas for the images of these modules in the algebras of quasisymmetric functions and peak functions under the relevant characteristic map. As initial applications, we resolve a question of Jing and Li (2015), introduce a new basis of the peak algebra analogous to the quasisymmetric Schur functions, uncover a new connection between Schur $Q$-functions and quasisymmetric Schur functions, give a representation-theoretic in…

关于对称化形式和 cyclotomic Hecke-Clifford 代数的 Schur 元素

On (super)symmetrizing forms and Schur elements of cyclotomic Hecke-Clifford algebras

本文引入了超对称化超代数的 Schur 元素，证明了 cyclotomic Hecke-Clifford 代数在特定条件下具有超对称性或对称性，并计算了半单情况下的 Schur 元素，同时定义了新的对称化形式。

Generalized q-Schur algebras and modular representation theory of finite groups with split (BN)-pairs

We introduce a generalized version of a q-Schur algebra (of parabolic type) for arbitrary Hecke algebras over extended Weyl groups. We describe how the decomposition matrix of a finite group with split BN-pair, with respect to a non-describing prime, can be partially described by the decomposition matrices of suitably chosen q-Schur algebras. We show that the investigated structures occur naturally in finite groups of Lie type.

A super Frobenius formula for the characters of cyclotomic Hecke algebras

We prove a super Frobenius formula for the characters of the cyclotomic Hecke algebras by applying the super Schur-Weyl reciprocity between the quantum superalgebras and cyclotomic Hecke algebras, which is a super analogue of the Frobenius formula in (T. Shoji, J. Algebra 226: 2000, 818--856) and a cyclotomic analogue of the super Frobeinus formula in (H. Mitsuhashi, Linear Multilinear Algebra 58: 2010, 941--955).

A categorical approach to classical and quantum Schur-Weyl duality

We use category theory to propose a unified approach to the Schur-Weyl dualities involving the general linear Lie algebras, their polynomial extensions and associated quantum deformations. We define multiplicative sequences of algebras exemplified by the sequence of group algebras of the symmetric groups and use them to introduce associated monoidal categories. Universal properties of these categories lead to uniform constructions of the Drinfeld functor connecting representation theories of the degenerate affine Hecke algebras and the Yangians and of its q-analogue. Moreover, we construct act…

渐近 Schur 代数与 q-Schur 代数的胞腔性

Asymptotic Schur algebras and cellularity of q-Schur algebras

本文证明了有限型 q-Schur 代数的胞腔性，推广了 Geck 的定理，并研究了渐近 Schur 代数的特殊模与 Schur 代数的左胞腔表示，推广了 Lusztig 的相关工作。