Description of the center of the affine Temperley-Lieb algebra of type $\widetilde{A_N}$

Type $\tilde{C}$ Temperley-Lieb algebra quotients and Catalan combinatorics

We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.

Framization of a Temperley-Lieb algebra of type $\mathtt{B}$

We extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$ and prove that such an extension supports a unique linear Markov trace function. We then introduce the Framization of the Temperley-Lieb algebra of type $\mathtt{B}$ as a quotient of the Yokonuma-Hecke algebra of type $\mathtt{B}$. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra of type $\mathtt{B}$ to pass to th…

Constructing a free resolution and remarks on bar homology of Temperley-Lieb algebra $TL_3$

In this paper I try to construct the Anick's resolution for the Temperley-Lieb algebra $TL_3$ and bar homology $Tor_\ast^{TL_3}(\mathbb{C},\mathbb{C})$ and shows that how $τ\in \mathbb{C}$ given in the definition of $TL_3$ is actively involved in the differentials both in free resolution and bar homology. It suggests that the bar homology depends on the nature of $τ$.

Gröbner-Shirshov bases for Temperley-Lieb algebras of the complex reflection group of type $G(d,1,n)$

We construct a Gröbner-Shirshov basis of the Temperley-Lieb algebra $\mathfrak{T}(d,n)$ of the complex reflection group $G(d,1,n)$, inducing the standard monomials expressed by the generators $\{E_i\}$ of $\mathfrak{T}(d,n)$. This result generalizes the one for the Coxeter group of type $B_n$ in \cite{KimSSLeeDI}. We also give a combinatorial interpretation of the standard monomials of $\mathfrak{T}(d,n)$, relating to the fully commutative elements of the complex reflection group $G(d,1,n)$. In this way, we obtain the dimension formula of $\mathfrak{T}(d,n)$.

A relation for the Jones-Wenzl projector and tensor space representations of the Temperley-Lieb algebra

A relation for the Jones-Wenzl projector is proven. It has the following consequence for representations of the Temperley-Lieb algebra on tensor product spaces: if such a representation is built from a Hermitian $n \times n$ matrix $T$ of rank $r$ such that $T^2=Q T$, then either $n^2 = Q^2 r$ and $Q^2 =1,2,3$ or $n^2 \geq 4 r$. For the latter class of representations, new examples are found. This includes explicit examples for $r=2,3,4$ and any $n \geq r$ (with one exception) and a solution for $n=r+1$ with arbitrary $r$.

Temperley-Lieb 晶体

Temperley-Lieb Crystals

本文针对 Temperley-Lieb 子式推广 Littlewood-Richardson 法则，引入洗牌 tableau，并利用 Stembridge 公理证明其构成 A 型 Kashiwara 晶体，从而给出相关 Schur 系数的组合解释。

Algebraic Systems for DNA Origami Motivated from Temperley-Lieb Algebras

We initiate an algebraic approach to study DNA origami structures by associating an element from a monoid to each structure. We identify two types of basic building blocks and describe an DNA origami structure with their composition. These building blocks are taken as generators of a monoid, called origami monoid, and, motivated by the well studied Temperley-Lieb algebras, we identify a set of relations that characterize the origami monoid. We also present several observations about the Green's relations for the origami monoid and study the relations to a cross product of Jones monoids that is…

Coxeter-Catalan combinatorics and Temperley-Lieb algebras

We introduce bijections between generalized type $A_n$ noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley-Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley-Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on $n+1$ strands. We then show that our b…

Degenerate 0-Schur algebras and Nil-Temperley-Lieb algebras

In \cite{JS} Jensen and Su constructed 0-Schur algebras on double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver approach naturally gives rise to a new class of algebras. That is, the path algebras defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters $\ut$. In particular, when all the entries of $\ut$ are 1, we have 0-Schur algerbas. When all the entries of $\ut$ are zero, we obtain a class of degenerate 0-Schur algebras. We prove that…

The Bubble Algebra: Structure of a Two-Colour Temperley-Lieb Algebra

We define new diagram algebras providing a sequence of multiparameter generalisations of the Temperley-Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional Statistical Mechanics. These algebras give a rigorous foundation to the various "multi-colour algebras" of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to…

Temperley-Lieb代数的拟遗传覆盖与相对控制维数

Quasi-hereditary covers of Temperley-Lieb algebras and relative dominant dimension

本文研究Temperley-Lieb代数的拟遗传覆盖问题，通过证明拟遗传代数相对控制维数的两倍关系，应用于Schur代数确定其相对控制维数，得到q-模拟及Hemmer-Nakano型结果，首次完整分类相关联系并确定整系数最佳拟遗传覆盖。

Temperley-Lieb代数的拓扑作用与表示稳定性

Topological actions of Temperley-Lieb algebras and Representation Stability

本文研究δ=1时Temperley-Lieb代数作为幺半群在拓扑空间上的作用，利用自然滤过推导同调群的表示结构，并证明滤过稳定的TL_n-空间序列可导出表示稳定的同调群序列，从而建立拓扑稳定性与表示稳定性之间的联系。