q-rook monoid algebras, Hecke algebras and Schur-Weyl duality

Presentations for rook partition monoids and algebras and their singular ideals

We obtain several presentations by generators and relations for the rook partition monoids and algebras, as well as their singular ideals. Among other results, we also calculate the minimal sizes of generating sets (some of our presentations use such minimal-size generating sets), and show that the singular part of the rook partition monoid is generated by its idempotents.

Jucys-Murphy elements of partition algebras for the rook monoid

Kudryavtseva and Mazorchuk exhibited Schur-Weyl duality between the rook monoid algebra $\mathbb{C}R_n$ and the subalgebra $\mathbb{C}I_k$ of the partition algebra $\mathbb{C} A_k(n)$ acting on $(\mathbb{C}^n)^{\otimes k}$. In this paper, we consider a subalgebra $\mathbb{C}I_{k+\frac{1}{2}}$ of $\mathbb{C} I_{k+1}$ such that there is Schur-Weyl duality between the actions of $\mathbb{C} R_{n-1}$ and $\mathbb{C} I_{k+\frac{1}{2}}$ on $(\mathbb{C}^n)^{\otimes k}$. This paper studies the representation theory of partition algebras $\mathbb{C}I_k$ and $\mathbb{C}I_{k+\frac{1}{2}}$ for rook monoid…

On Tensor Spaces for Rook Monoid Algebras

Let $m,n\in \mathbb{N}$, and $V$ be a $m$-dimensional vector space over a field $F$ of characteristic $0$. Let $U=F\oplus V$ and $R_n$ be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of $U^{\otimes n}$ in $FR_n$, which comes from some one-dimensional two-sided ideal of rook monoid algebra. We show that the two-sided ideal generated by this element is indeed the whole annihilator of $U^{\otimes n}$ in $FR_n$.

Presentation of the Motzkin Monoid

In 2010, Tom Halverson and Georgia Benkart introduced the Motzkin algebra, a generalization of the Temperley-Lieb algebra, whose elements are diagrams that can be multiplied by stacking one on top of the other. Halverson and Benkart gave a diagrammatic algorithm for decomposing any Motzkin diagram into diagrams of three subalgebras: the Right Planar Rook algebra, the Temperley-Lieb algebra, and the Left Planar Rook algebra. We first explore the Right and Left Planar Rook monoids, by finding presentations for these monoids by generators and relations, using a counting argument to prove that our…

Rook 幺半群的快速傅里叶变换

Fast Fourier Transforms for the Rook Monoid

本文定义了 Rook 幺半群（也称为对称逆半群）的傅里叶变换概念，并提出了两种高效的分治算法（快速傅里叶变换，FFT）来计算它，首次将群 FFT 扩展到非群半群。

Characters of Renner Monoids and Their Hecke Algebras

This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e., the Cartan type $A$ Renner monoid) was computed earlier by Dieng, Halverson, and Poladian using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type $C$ and use them to obtain an explicit descript…

不可约特征标与q-罗克单形的位迹

Irreducible characters and bitrace for the $q$-rook monoid

本文通过顶点代数方法研究q-罗克单形代数的不可约特征标，推导出新的特征标公式，并证明了穆纳根-纳卡亚马规则，同时引入了q-罗克单形的位迹并推导其组合公式。

作为车 monoid 表示中心化子的半单划分代数

Semi-simple partition algebras as centralizers of representations for rook monoids

本文证明了半单划分代数可作为车 monoid 平凡表示的迭代限制-诱导表示的中心化子代数，并给出了该表示到张量空间的分解及由广义 Bell 数给出的维数。

Effective representations of Hecke-Kiselman monoids of type A

We prove effectiveness of certain representations of Hecke-Kiselman monoids of type A constructed by Ganyushkin and Mazorchuk and also construct further classes of effective representations for these monoids. As a consequence the effective dimension of monoids of type A is determined. We also show that odd Fibonacci numbers appear as the cardinality of certain bipartite HK-monoids and count the number of multiplicity free elements in any HK-monoid of type A.

仿射图范畴、代数和单体

Affine diagram categories, algebras and monoids

本文研究了仿射图代数的生成元和关系，探讨了其表示论及张量积表示的渐进行为，并验证了一个关于有限单体表示中不可分解因子数量的猜想。

Rook 特征标作为对称函数

Rook characters as symmetric functions

本文刻画了对称函数环中一个非齐次基，其求值对应 rook 幺半群不可约表示特征标，并利用 Schur–Weyl 对偶揭示其与传播分拆代数及 Schur 函数 smash 积的关联。

An action of the Hecke monoid on rational modules for the Borel subgroup of a quantised general linear group

We construct an action of the Hecke monoid on the category of rational modules for the quantum negative Borel subgroup of the quantum general linear group. We also show that this action restricts to the category of polynomial modules for this quantum subgroup and induces an action on the category of modules for quantised Borel-Schur algebras.

Gelfand Models for Diagram Algebras

A Gelfand model for a semisimple algebra A over C is a complex linear representation that contains each irreducible representation of A with multiplicity exactly one. We give a method of constructing these models that works uniformly for a large class of semisimple, combinatorial diagram algebras including: the partition, Brauer, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, and planar rook monoid algebras. In each case, the model representation is given by diagrams acting via "signed conjugation" on the linear span of their vertically symmetric diagrams. This representation is a generali…

On monoidal Koszul duality for the Hecke category

We attempt to give a gentle (though ahistorical) introduction to Koszul duality phenomena for the Hecke category, focusing on the form of this duality studied in joint work of Achar, Riche, Williamson, and the author. We illustrate some key phenomena and constructions for the simplest nontrivial case of (finite) $\mathrm{SL}_2$ using Soergel bimodules, a concrete algebraic model of the Hecke category.

Representations of the Rook-Brauer Algebra

We study the representation theory of the rook-Brauer algebra RB_k(x), also called the partial Brauer algebra. This algebra has a basis of "rook-Brauer" diagrams, which are Brauer diagrams that allow for the possibility of missing edges. The Brauer, Temperley-Lieb, Motzkin, rook monoid, and symmetric group algebras are all subalgebras of the rook-Brauer algebra. We prove that RB_k(n) is the centralizer algebra of the complex orthogonal group O(n) acting on the k-fold tensor power of the sum of its 1-dimensional trivial module and its n-dimensional defining module, and thus the rook-Brauer alge…

Artin group injection in the Hecke algebra for right-angled groups

We prove some injectivity results: that a Coxeter monoid $\mathbb{Z}$-algebra (or $0$-Hecke algebra) injects in the incidence $\mathbb{Z}$-algebra of the corresponding Bruhat poset, for any Coxeter group; that the Hecke algebra of a right-angled Coxeter group injects in the Coxeter monoid $\mathbb{Z}[q,q^{-1}]$-algebra (and then in the incidence $\mathbb{Z}[q,q^{-1}]$-algebra of the corresponding Bruhat poset); that a right-angled Artin group injects in the group of invertible elements of the Hecke algebra of the corresponding Coxeter group (and then in the group of invertible elements of a Co…

Generic Hecke algebra for Renner monoids

We associate with every Renner monoid $R$ a \emph{generic Hecke algebra} $\H(R)$ over $\mathbb{Z}[q]$ which is a deformation of the monoid $\mathbb{Z}$-algebra of $R$. If $M$ is a finite reductive monoid with Borel subgroup $B$ and associated Renner monoid $R$, then we obtain the associated Iwahori-Hecke algebra $\H(M,B)$ by specialising $q$ in $\H(R)$ and tensoring by $\mathbb{C}$ over $\mathbb{Z}$, as in the classical case of finite algebraic groups. This answers positively to a long-standing question of L. Solomon.

Hecke与Artin幺半群及其同态

Hecke and Artin monoids and their homomorphisms

本文系统研究Hecke和Artin幺半群的同态，发现抛物线投影等保持抛物线元素的同态可提升至覆盖Artin幺半群，并揭示了一个包含多数新同态的更大范畴。

表格与有限变换单形的轨道调和商

Tableaux and orbit harmonics quotients for finite transformation monoids

本文提出了一种特征无关的方法，用于研究有限变换单形的轨道调和商的分层模块结构，建立了两个分支规则，并给出了多项式环的Cauchy分解的类比。

Dominance order and monoidal categorification of cluster algebras

We study a compatibility relationship between Qin's dominance order on a cluster algebra $\mathcal{A}$ and partial orderings arising from classifications of simple objects in a monoidal categorification $\mathcal{C}$ of $\mathcal{A}$. Our motivating example is Hernandez-Leclerc's monoidal categorification using representations of quantum affine algebras. In the framework of Kang-Kashiwara-Kim-Oh's monoidal categorification via representations of quiver Hecke algebras, we focus on the case of the category $R-gmod$ for a symmetric finite type $A$ quiver Hecke algebra using Kleshchev-Ram's classi…

Hecke单群、其同态与抛物性

Hecke monoids, their homomorphisms and parabolicity

研究Hecke单群的同态，特别是映射抛物元到抛物元的同态，以及其注入性。通过分类局部注入的连通同态，揭示了Hecke单群同态的结构。

p-进Iwahori Hecke代数与对偶Vinberg单形

Pro-p Iwahori Hecke algebras and the dual Vinberg monoid

本文研究了p-进Iwahori Hecke代数与对偶Vinberg单形的关系，并在GL(n)情况下推广了SpecZ由半单表示参数化的结果。

Irreducible representations of Hecke-Kiselman monoids

Let $K[HK_Θ]$ denote the Hecke-Kiselman algebra of a finite oriented graph $Θ$ over an algebraically closed field $K$. All irreducible representations, and the corresponding maximal ideals of $K[HK_Θ]$, are characterized in case this algebra satisfies a polynomial identity. The latter condition corresponds to a simple condition that can be expressed in terms of the graph $Θ$. The result shows a surprising similarity to the classical results on representations of finite semigroups; namely every representation either comes form an idempotent in the Hecke-Kiselman monoid $HK_Θ$ (and hence it is $…

Set-partition tableaux and representations of diagram algebras

The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis ind…

A diamond lemma for Hecke-type algebras

In this paper we give a version of Bergman's diamond lemma which applies to certain monoidal categories presented by generators and relations. In particular, it applies to: the Coxeter presentation of the symmetric groups, the quiver Hecke algebras of Khovanov-Lauda-Rouquier, the Webster tensor product algebras, and various generalizations of these. We also give an extension of Manin-Schechtmann theory to non-reduced expressions.

On some actions of the 0-Hecke monoids of affine symmetric groups

There are left and right actions of the 0-Hecke monoid of the affine symmetric group $\tilde{S}_n$ on involutions whose cycles are labeled periodically by nonnegative integers. Using these actions we construct two bijections, which are length-preserving in an appropriate sense, from the set of involutions in $\tilde{S}_n$ to the set of $\mathbb{N}$-weighted matchings in the $n$-element cycle graph. As an application, we compute a formula for the bivariate generating function counting the involutions in $\tilde S_n$ by length and absolute length. The 0-Hecke monoid of $\tilde{S}_n$ also acts on…

党-海克斯代数

Party-Hecke algebras

本文引入了双参数变形的党代数，研究其基和作为编织与系代数的商代数的实现，探讨党独占体与系对称体的关系。

Dual braid monoids, Mikado braids and positivity in Hecke algebras

We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written $x^{-1} y$ where $x$ and $y$ are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type $A_n$ and $B_n$ and then show that in spherical types different from $D_n$ the simple elements of the dual braid monoid (for arbitrary choice of Coxeter element) embedded in the braid group are rational permutation braids (we conjecture this to hold also in type $D_n$).This property implies positivity properties of the polynom…

Ext-enhanced monoidal Koszul duality for $\mathrm{GL}_2$

The Hecke category participates in an equivalence called monoidal Koszul duality, which exchanges it with the category of (Langlands-dual) "free-monodromic tilting sheaves." Motivated by a recent conjecture of Gorsky and the first-named author on HOMFLYPT link homology, we propose to enhance this duality with an additional grading. We provide evidence for this enhancement in the case of $\mathrm{GL}_2$, working in the language of the second-named author's joint work with Achar, Riche, and Williamson.

A subalgebra of 0-Hecke algebra

Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on the set of all subsets of $I$ using unipotent $χ$-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.

箭图Hecke代数上的$\boldsymbol{\frak{C}_{wv}}$范畴的幺子种子

Monoidal seeds of the categories $\mathcal{C}_{wv}$ over quiver Hecke algebras

针对对称箭图Hecke代数，利用反射函子与新算子构造了$\boldsymbol{\frak{C}_{wv}}$的量子幺子种子，证明其与特定行列式模的素因子集合一致，且$\boldsymbol{\frak{C}_{wv}}$的Grothendieck环介于簇代数与上簇代数之间。

The biHecke monoid of a finite Coxeter group

The usual combinatorial model for the 0-Hecke algebra of the symmetric group is to consider the algebra (or monoid) generated by the bubble sort operators. This construction generalizes to any finite Coxeter group W. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it has |W| simple and projective modules. In order to construct a combinatorial model for the simp…

Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

We associate a monoidal category $\mathcal{H}^λ$ to each dominant integral weight $λ$ of $\widehat{\mathfrak{sl}}_p$ or $\mathfrak{sl}_\infty$. These categories, defined in terms of planar diagrams, act naturally on categories of modules for the degenerate cyclotomic Hecke algebras associated to $λ$. We show that, in the $\mathfrak{sl}_\infty$ case, the level $d$ Heisenberg algebra embeds into the Grothendieck ring of $\mathcal{H}^λ$, where $d$ is the level of $λ$. The categories $\mathcal{H}^λ$ can be viewed as a graphical calculus describing induction and restriction functors between categor…

Coherent categorification of quantum loop algebras : the $SL(2)$ case

We construct an equivalence of graded Abelian categories from a category of representations of the quiver-Hecke algebra of type $A_1^{(1)}$ to the category of equivariant perverse coherent sheaves on the nilpotent cone of type $A$. We prove that this equivalence is weakly monoidal. This gives a representation-theoretic categorification of the preprojective K-theoretic Hall algebra considered by Schiffmann-Vasserot. Using this categorification, we compare the monoidal categorification of the quantum open unipotent cells of type $A_1^{(1)}$ given by Kang-Kashiwara-Kim-Oh-Park in terms of quiver-…

Double Catalan monoids

In this paper we define and study what we call the double Catalan monoid. This monoid is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations. We show that the double Catalan monoid provides an algebraization of the (combinatorial) set of 4321-avoiding permutations and relate its combinatorics to various off-shoots of both the combinatorics of Catalan numbers and the combinatorics of permutations. In particular, we give an algebraic interpretation of the first derivative of the Kreweras involution on Dyck paths, of 4321-avoiding involutions and of recent results…

Motzkin monoids and partial Brauer monoids

We study the partial Brauer monoid and its planar submonoid, the Motzkin monoid. We conduct a thorough investigation of the structure of both monoids, providing information on normal forms, Green's relations, regularity, ideals, idempotent generation, minimal (idempotent) generating sets, and so on. We obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent-generated. We find formulae for the rank (smallest size of a generating set) of each ideal, and for the idempotent rank (smallest size of an idempotent generating set) of the idempotent-generated su…

On the representation theory of finite J-trivial monoids

In 1979, Norton showed that the representation theory of the 0-Hecke algebra admits a rich combinatorial description. Her constructions rely heavily on some triangularity property of the product, but do not use explicitly that the 0-Hecke algebra is a monoid algebra. The thesis of this paper is that considering the general setting of monoids admitting such a triangularity, namely J-trivial monoids, sheds further light on the topic. This is a step to use representation theory to automatically extract combinatorial structures from (monoid) algebras, often in the form of posets and lattices, both…

Localizations for quiver Hecke algebras II

We prove that the localization of the monoidal category $\mathcal{C}_w$ is rigid, and the category $\mathcal{C}_{w,v}$ admits a localization via a real commuting family of central objects. Note that the localization of $\mathcal{C}_{w,v}$ categorifies the open Richardson variety.

Localizations for quiver Hecke algebras

We provide the localization procedure for monoidal categories by a real commuting family of braiders. For an element $w$ of the Weyl group, $\mathscr{C}_w$ is a subcategory of modules over quiver Hecke algebra which categorifies the quantum unipotent coordinate algebra $A_q[\mathfrak{n}(w)]$. We construct the localization $\widetilde{\mathscr{C}_w}$ of $\mathscr{C}_w$ by adding the inverses of simple modules which correspond to the frozen variables in the quantum cluster algebra $A_q[\mathfrak{n}(w)]$. The localization $\widetilde{\mathscr{C}_w}$ is left rigid and we expect that it is rigid.

The Character Theory of a Complex Group

We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. Our focus is the Hecke category of Borel-equivariant D-modules on the flag variety of a complex reductive group G (equivalently, the category of Harish Chandra bimodules of trivial central character) and its monodromic variant. The Hecke category is a categorified analogue of the finite Hecke algebra, which is a finite-dimensional semi-simple symmetric Frobenius algebra. We establish parallel properties of the Hecke category, showing it is a two-dualizable Calabi-Yau …

Singular Hecke algebras, Markov traces, and HOMFLY-type invariants

We define the singular Hecke algebra ${\mathcal H} (SB_n)$ as the quotient of the singular braid monoid algebra ${\mathbb C} (q) [SB_n]$ by the Hecke relations $σ_k^2 = (q-1) σ_k +q$, $1 \le k\le n-1$, and define the Markov traces on the sequence $\{{\mathcal H}(SB_n)\}_{n=1}^{+\infty}$ in the same way as for the Markov traces on the tower of (non-singular) Hecke algebras of the symmetric groups. We prove that the Markov traces are in one-to-one correspondance with the invariants that satisfies some skein relation, and compute an explicit classification of the Markov traces. Thanks to this cla…

关于仿射 Hecke 代数的两个模几何实现

On two modular geometric realizations of an affine Hecke algebra

本文通过构造三类范畴的幺半等价，将正特征域上仿射 Hecke 代数与 Harish-Chandra 双模、Steinberg 簇等变凝聚层及仿射旗簇构造层联系起来，并据此证明了 Finkelberg-Mirković 关于主块几何实现的猜想。

Newton-Okounkov bodies for categories of modules over quiver Hecke algebras

We show that for a finite-type Lie algebra $\mathfrak{g}$, the representation theory of quiver Hecke algebras provides a natural framework for the construction of Newton-Okounkov bodies associated to the quantum coordinate rings $\Aqnw$. When $\mathfrak{g}$ is simply-laced, we use Kang-Kashiwara-Kim-Oh's monoidal categorification to investigate the cluster theory of these bodies. In particular, our construction yields a simplex $\ds$ for every seed $\s$ of $\Aqnw$. We exhibit various properties of these simplices by characterizing their rational points, normal fans, and volumes. As an applicat…

Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras

We study the representation theory of three towers of algebras which are related to the symmetric groups and their Hecke algebras. The first one is constructed as the algebras generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. The two others are the monoid algebras of nondecreasing functions and nondecreasing parking functions. For these three towers, we describe the structure of simple and indecomposable projective modules, together with the Cartan map. The Grothendieck algebras and coalgebras given respectively by the induct…

The lattice of nil-Hecke algebras over real and complex reflection groups

Associated to every complex reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, and which are obtained by killing all braid words that are "sufficiently long", as well as some integer power of each generator. These include usual nil-Coxeter algebras, nil-Temperley-Lieb algebras, and their variants, and lead to symmetric semigroup module categories which necessarily cannot be monoidal. Motivated by classical work of Coxeter (1957) and the Broue-Malle-Rouquier freeness conjecture [Crelle 1998], and continuing beyond work of the sec…

dg-Hecke 对偶性与张量积

dg-Hecke duality and tensor products

本文继续研究幺半范畴 $D(G)$，将上同调对偶函子转移到 Hecke dg-模导出范畴，发展更一般的反对合方法，并验证张量积对应 dg-侧的 operadic 张量积。

A Survey on Springer Theory

This is not standard in the sense that we understand a Springer map to be a collapsing of homogeneous bundles. Apart from that we use mostly techniques from Chriss and Ginzbergs book but we work in the equivariant derived category of Bernstein and Lunts. We define Steinberg algebras as (equivariant) Borel-Moore homology algebras of the associated Steinberg varieties. The data of the BBD-decomposition theorem applied to the Springer map give a parametrization of projective graded and simple graded modules over the Steinberg algebra. Also, the projective graded modules are equivalent to a catego…

Finitary 2-categories associated with dual projection functors

We study finitary 2-categories associated to dual projection functors for finite dimensional associative algebras. In the case of path algebras of admissible tree quivers (which includes all Dynkin quivers of type A) we show that the monoid generated by dual projection functors is the Hecke-Kiselman monoid of the underlying quiver and also obtain a presentation for the monoid of indecomposable subbimodules of the identity bimodule.

The Hecke group algebra of a Coxeter group and its representation theory

Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, ...). In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementa…

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。

Categorification of the adjoint action of quantum groups

Let $U$ be a quantized enveloping algebra. We consider the adjoint action of an $\mathfrak{sl}_2$-subalgebra of $U$ on a subalgebra of $U^+$ that is maximal integrable for this action. We categorify this representation in the context of quiver Hecke algebras. We obtain an action of the 2-category associated with $\mathfrak{sl}_2$ on a category of modules over certain quotients of quiver Hecke algebras. Our approach is similar to that of Kang-Kashiwara for categorifications of highest weight modules via cyclotomic quiver Hecke algebras. One of the main new features is a compatibility of the cat…

Laurent phenomenon and simple modules of quiver Hecke algebras

We study consequences of a monoidal categorification of the unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category $\mathcal C_w$ strongly commutes with all the cluster variables in a cluster $[ \mathscr C]$, then $[S]$ is a cluster monomial in $[ \mathscr C ]$. If $S$ strongly commutes with cluster variables except exactly one cluster variable $[M_k]$, then $[S]$ is either a cluster monomial in $[\mathscr C ]$ or a cluster monomial in $μ_k([ \mathscr C ])$. We give a new proof of th…

Tied–boxed 代数

Tied--boxed algebras

本文定义 tied–boxed Hecke 与 tied–boxed Temperley–Lieb 两种新代数，研究其表示论与胞腔基，并通过框式分支幺半群建立其图解解释及与 partition Temperley–Lieb 代数的联系。

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…