S. R. Doty, R. M. Green
We obtain a presentation of certain affine q-Schur algebras in terms of generators and relations. The presentation is obtained by adding more relations to the usual presentation of the quantized enveloping algebra of type affine gl_n. Our results extend and rely on the corresponding result for the q-Schur algebra of the symmetric group, which were proved by the first author and Giaquinto.
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We give a presentation of cyclotomic q-Schur algebras by generators and defining relations. As an application, we give an algorithm for computing decomposition numbers of cyclotomic q-Schur algebras.
We prove that the double affine Hecke algebra of type A is Morita equivalent to the quantized affine Schur algebra.
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.
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…
The affine quantum Schur algebra is a certain important infinite dimensional algebra whose representation theory is closely related to that of quantum affine $\frak{gl}_n$. Finite dimensional irreducible modules for the affine quantum Schur algebra ${\mathcal S}_{\vartriangle}(n,r)_{v}$ were classified in \cite{DDF}, where $v\in{\mathbb C}^*$ is not a root of unity. We will classify blocks of the affine quantum Schur algebra ${\mathcal S}_{\vartriangle}(n,r)_{v}$ in this paper.
We will classify finite dimensional irreducible modules for affine quantum Schur algebras at roots of unity and generalize \cite[(6.5f) and (6.5g)]{Gr80} to the affine case in this paper.
When the parameter $q\in\mathbb C^*$ is not a root of unity, simple modules of affine $q$-Schur algebras have been classified in terms of Frenkel--Mukhin's dominant Drinfeld polynomials (\cite[4.6.8]{DDF}). We compute these Drinfeld polynomials associated with the simple modules of an affine $q$-Schur algebra which come from the simple modules of the corresponding $q$-Schur algebra via the evaluation maps.
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 and cyclotomic $q$-Schur categories via webs
本文通过构造仿射 q-web 范畴、仿射 q-Schur 范畴及其分圆商范畴并建立图解整基，首次给出仿射与分圆 q-Schur 代数的图解表示。
Associated to the two types of finite dimensional simple superalgebras, there are the general linear Lie superalgebra and the queer Lie superalgebra. The universal enveloping algebras of these Lie superalgebras act on the tensor spaces of the natural representations and, thus, define certain finite dimensional quotients, the Schur superalgebras and the queer Schur superalgebra. In this paper, we introduce the quantum analogue of the queer Schur superalgebra and investigate the presentation problem for both the queer Schur superalgebra and its quantum analogue.
Affine and cyclotomic Schur categories
利用仿射编织范畴构造仿射Schur范畴和分圆Schur范畴，给出其图式基和双SST基，并证明与退化分圆Schur代数的同构。
K-theoretic construction of affine quantum Schur algebras and iquantum groups with unequal parameters
利用Kato的奇异框架，通过等变K理论构造了三参数C型仿射量子Schur代数，进而实现三参数AIII型仿射i量子群的等变K理论实现。
Affine quantum Schur algebras and $\imath$quantum groups with three parameters
本文研究三参数 C 型仿射 Hecke 代数对应的仿射量子 Schur 代数，推导半单生成元乘法公式并利用 BLM 稳定化构造三参数仿射 AIII 型拟分裂 $\imath$ 量子群。
We obtain a presentation by generators and relations for generalized Schur algebras and their quantizations. This extends earlier results obtained in the type A case. The presentation is compatible with Lusztig's modified form of a quantized enveloping algebra. We show that generalized Schur algebras inherit a canonical basis from the modified form of the quantized enveloping algebra, that this gives them a cellular structure, and thus they are quasihereditary over a field.
Let ${\mathsf F}$ be the Schur functor from the category of finite dimensional ${\mathcal H}_{\vartriangle}(r)_\mathbb C$-modules to the category of finite dimensional ${\mathcal S}_{\vartriangle}(n,r)_{\mathbb{C}}$-modules, where ${\mathcal H}_{\vartriangle}(r)_\mathbb C$ is the extended affine Hecke algebra of type $A$ over ${\mathbb C}$ and ${\mathcal S}_{\vartriangle}(n,r)_{\mathbb{C}}$ is the affine quantum Schur algebras over $\mathbb{C}$. The Drinfeld polynomials associated with ${\mathsf F}(V)$ were determined in \cite[7.6]{CP96} and \cite[4.4.2]{DDF} in the case of $n>r$, where $V$ is…
Let ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the double Ringel--Hall algebra of the cyclic quiver $\triangle(n)$ and let $\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the modified quantum affine algebra of ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$. We will construct an integral form $\dot{{\mathfrak D}_{\vartriangle}}(n)$ for $\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ such that the natural algebra homomorphism from $\dot{{\mathfrak D}_{\vartriangle}}(n)$ to the integral affine quantum Schur algebra is surjective. Furthermore, we will use Hall algebras to construct…
For a Dynkin quiver $Q$ of type ADE and a sum $β$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of the corresponding type via equivariant K-theory, imitating Ginzburg-Reshetikhin-Vasserot's geometric realization of the quantum affine Schur-Weyl duality. Our construction is based on Hernandez-Leclerc's isomorphism between a certain graded quiver variety and the space of representations of the quiver $Q$ of dimension vector $β$. We identify the functor induced from our bimodule with Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl d…
We investigate the structure of the double Ringel-Hall algebras associated with cyclic quivers and its connections with quantum loop algebras of $\mathfrak{gl}_n$, affine quantum Schur algebras and affine Hecke algebras. This includes their Drinfeld-Jimbo type presentation, affine quantum Schur-Weyl reciprocity, representations of affine quantum Schur algebras, and connections with various existing works by Lusztig, Varagnolo-Vasserot, Schiffmann, Hubery, Chari-Pressley, Frenkel-Mukhin, etc. We will also discuss conjectures on a realization of Beilinson-Lusztig-MacPherson type and Lusztig type…
For a Dynkin quiver $Q$ (of type ADE), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc's monoidal category $\mathcal{C}_Q$ of modules over the quantum loop algebra $U_q(L\mathfrak{g})$ via Nakajima's homomorphism. As an application, we show that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives an equivalence between the cate…
Schur-Weyl dualities for shifted quantum affine algebras and Ariki-Koike algebras
建立移位量子仿射代数与Ariki-Koike代数间的Schur-Weyl对偶，并实现分圆q-Schur代数。
In this paper we prove that the affine Schur algebra $\whS(n,r)$ is affine quasi-hereditary. This result is then applied to show that $\whS(n,r)$ has finite global dimension and its centralizer subquotient algebras are Laurent polynomial algebras. We also use the result to give a parameter set of simple $\whS(n,r)$-modules and identify this parameter set with that given in \cite{DDF}.
We categorify the extended affine Hecke algebra and the affine quantum Schur algebra S(n,r) for 2 < r < n, using Elias-Khovanov and Khovanov-Lauda type diagrams. We also define the affine analogue of the Elias-Khovanov and the Khovanov-Lauda 2-representations of these categorifications into an extension of the 2-category of affine (singular) Soergel bimodules.
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。
We introduce the notion of standard multipartitions and establish a one-to-one correspondence between standard multipartitions and irreducible representations with integral weights for the affine Hecke algebra of type A with a parameter q which is not a root of unity. We then extend the correspondence to all Kleshchev multipartitions for Ariki-Koike algebras of integral type. By the affine Schur--Weyl duality, we further extend this to a correspondence between standard multipartitions and Drinfeld multipolynomials of integral type whose associated irreducible polynomial representations complet…
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…
This is a follow-up to the paper in which we categorified the affine quantum Schur algebra S(n,r) for 2 < r < n, using a quotient of Khovanov and Lauda's categorification of the affine quantum sl_n. In this paper we categorify S(n,n) for n > 2, using an extension of the aforementioned quotient.
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。
Geometric construction of Schur algebras
本文通过广义Steinberg簇的Borel-Moore同调与等变K群构造了任意型的广义Schur代数，并以此建立了局部几何Langlands对应的类似物、仿射AIII型拟分裂$\imath$量子群的实现及几何Howe对偶。
Gelfand--Graev functor and quantum affine Schur--Weyl duality
本文揭示了中心覆盖的Gelfand--Graev模、Arnold--Brieskorn流形的Euler--Poincaré多项式与量子仿射Schur--Weyl对偶之间的联系，证明Gelfand--Graev函子与该对偶相关，且其Iwahori不动部分的交换代数是量子群的商。
We first provide an explicit combinatorial description of the Auslander-Reiten quiver $Γ^Q$ of finite type $D$. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra $U_q'(D^{(i)}_{n+1})$ $(i=1,2)$ and the quiver Hecke algebra $R_{D_{n+1}}$ associated to $D_{n+1}$ $(n \ge 3)$, by using the combinatorial description and the generalized quantum affine Schur-Weyl duality functor. As applications, we can prove that Dorey's rule holds for the category $\Rep(R_{D_{n+1}})$ and prove an interesting difference between multiplicity free positive roo…
We establish an explicit embedding of a quantum affine $\mathfrak{sl}_n$ into a quantum affine $\mathfrak{sl}_{n+1}$. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine $\mathfrak{sl}_n$ and $\mathfrak{gl}_n$. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation…
Presenting mirabolic quantum Schur algebras $\mathcal{MS}_v(2,d)$
本文通过生成元和关系式给出了mirabolic量子Schur代数$\mathcal{MS}_v(2,d)$的显式呈现，确定了其与Rosso的mirabolic量子$\mathfrak{sl}_2$的商同构，并给出了分裂Wedderburn分解及权幂等元表示。
We study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal system for a fixed convex preorder. A cuspidal system consists of irreducible cuspidal modules---one for each real positive root for the corresponding affine root system ${\tt X}_l^{(1)}$, as well as irreducible imaginary modules---one for each $l$-multipartition. We study imaginary modules by means of `imaginary Schur-Weyl duality'. We introduce an imagina…
We introduce an analogue of the $q$-Schur algebra associated to Coxeter systems of type $\hat A_{n-1}$. We give two constructions of this algebra. The first construction realizes the algebra as a certain endomorphism algebra arising from an affine Hecke algebra of type $\hat A_{r-1}$, where $n \geq r$. This generalizes the original $q$-Schur algebra as defined by Dipper and James, and the new algebra contains the ordinary $q$-Schur algebra and the affine Hecke algebra as subalgebras. Using this approach we can prove a double centralizer property. The second construction realizes the affine $q$…
The quantum Frobenius map and it splitting are shown to descend to corresponding maps for generalized $q$-Schur algebras at a root of unity. We also define analogs of $q$-Schur algebras for any affine algebra, and prove the corresponding results for them.
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…
Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras
该文构造量子仿射对称对与定向 KLR 代数间的函子，推广广义 Schur-Weyl 对偶性，并恢复拟分裂 AIII 型的已知对偶结果。
We define a new family of symmetric functions which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions including symmetry, dominance and conjugation. We conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine Schur functions. As applications, we show how affine Stanley symmetric functions generalise the (dual of the) $k$-Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. Conjecturally, affine Stanley symmetric functions should be related to the coh…
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…
The Schur-Weyl duality, which started as the study of the commuting actions of the symmetric group $S_d$ and $\mathrm{GL}(n,\mathbb{C})$ on $V^{\otimes d}$ where $V=\mathbb{C}^n$, was extended by Drinfeld and Jimbo to the context of the finite Iwahori-Hecke algebra $H_d(q^2)$ and quantum algebras $U_q(\mathrm{gl}(n))$, on using universal $R$-matrices, which solve the Yang-Baxter equation. There were two extensions of this duality in the Hecke-quantum case: to the affine case, by Chari and Pressley, and to the super case, by Moon and by Mitsuhashi. We complete this chain of works by completing …
We will construct the Lusztig form for the quantum loop algebra of $\mathfrak{gl}_n$ by proving the conjecture \cite[3.8.6]{DDF} and establish partially the Schur--Weyl duality at the integral level in this case. We will also investigate the integral form of the modified quantum affine $\mathfrak{gl}_n$ by introducing an affine stabilisation property and will lift the canonical bases from affine quantum Schur algebras to a canonical basis for this integral form. As an application of our theory, we will also discuss the integral form of the modified extended quantum affine $\mathfrak{sl}_n$ and…
Schur-Weyl duality for quantum toroidal superalgebras
本文建立了双重仿射 Hecke 代数与量子环面超代数之间的 Schur-Weyl 型对偶，将 Vasserot-Varagnolo [VV96] 的经典结果推广到了超情形。
The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_λ^{(k)}$, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions and $k+1$-cores, we establish an algorithm for identifying chains in the $k$-Young l…
In this paper we prove that the affine Schur algebra $\s$ is affine cellular over $\bbq$. As an application, we prove it is of finite global dimension.
Affine quantum Schur--Weyl duality
本文证明了仿射量子Schur-Weyl对偶：在特征零且Hecke参数非单位根时，扩展仿射Hecke代数到仿射张量空间自同态代数的自然同态是同构，并由此证明了关于仿射量子Schur代数中心和Noether性的两个猜想。
Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$
本文为仿射超杨代数 $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$ 构造了 cut-and-join 算子及其本征函数超 Schur 函数，并推广了 Pieri 规则、hook 公式和 Cauchy 公式，同时探讨了其弦论起源。
By using methods developed in arXiv:math/0602181 we study the irreducibility of certain Wakimoto modules for $\widehat{sl_2}$ at the critical level. We classify all $χ\in {\Bbb C}((z))$ such that the corresponding Wakimoto module $W_χ$ is irreducible. It turns out that zeros of Schur polynomials play important rule in the classification result.
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。
Algebra of operators for Q-Schur polynomials
本研究从仿射Yangian视角，给出作用于Schur、Q-Schur多项式且对应KP和BKP可积族的算子代数，分析其交换子代数、本征值盒可加性及算子单钩展开性质。
Affine flag varieties and quantum symmetric pairs
本文通过仿射C型部分旗簇的Schur与Lusztig代数，构造了仿射型量子对称对的几何实现，并建立了相应量子代数的典范基及其正性性质。
Geometric Approach to I-Quantum Group of Affine Type D
本文采用几何方法研究仿射型D相关的Schur代数与Lusztig代数结构，证明Lusztig代数的一个子代数经稳定化得到仿射$\mathfrak{sl}$型量子群的余理想子代数，还构造了幂等量子代数的单项式基与典范基并确立其正性性质。
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.
We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the type-$A$ affine Weyl group and produce a crystal reflecting the internal structure of the generalized Young modules whose Frobenius image is represented by stable Schubert polynomials. We apply the crystal framework to products of a Schur function with a $k$-Schur function, consequently proving that a subclass of 3-point Gromov-Witten invariants of complete flag va…
We show that the defining relations needed to describe a generalized q-Schur algebra as a quotient of a quantized enveloping algebra are determined completely by the defining ideal of a certain finite affine variety, the points of which correspond bijectively to the set of weights. This explains, unifies, and extends previous results.
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…
A cuspidal system for an affine Khovanov-Lauda-Rouquier algerba $R_\al$ yields a theory of standard modules. This allows us to classify the irreducible modules over $R_\al$ up to the so-called imaginary modules. We make a conjecture on reductions modulo $p$ of irreducible $R_\al$-modules, which generalizes James Conjecture. We also describe minuscule imaginary modules, laying the groundwork for future study of imaginary Schur-Weyl duality. We introduce colored imaginary tensor spaces and reduce a classification of imaginary modules to one color. We study the characters of cuspidal modules. We …
Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $\mathcal{C}^0_{\mathfrak{g}}$ be Hernandez-Leclerc's category. For a duality datum $\mathcal{D}$ in $\mathcal{C}^0_{\mathfrak{g}}$, we denote by $\mathcal{F}_{\mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give sufficient conditions for a duality datum $\mathcal{D}$ to provide the functor $\mathcal{F}_{\mathcal{D}}$ sending simple modules to simple modules. Then we introduce the notion of cuspidal modules in $\mathcal{C}^0_{\mathfrak{g}}$, and show that all simple modules in $\mathcal{C}…
We give an explicit expression for the central elements of affine Hecke algebras of type A in the Coxeter presentation, in terms of (parabolic) affine Kazhdan-Lusztig polynomials. Our approach is based on a version of quantum affine Schur-Weyl duality involving the Hall algebra of a cyclic quiver, on the theory of canonical bases and on the higher-level q-deformed Fock spaces recently introduced by Uglov.
Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to whom they are related via Schur-Weyl duality. In this review paper, we give a glimpse on some aspects of their very rich representation theory in the context of general quantum affinizations. We illustrate with several examples. We also announce new results and explain possible further developments, in particular on finite dime…
The affine and degenerate affine Birman-Murakami-Wenzl (BMW) algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce acti…
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