Akihide Hanaki
We define modular Terwilliger algebras of association schemes, Terwilliger algebras over a positive characteristic field, and consider basic properties. We give a condition for the modular Terwilliger algebra to be non-semisimple. We show that the dimension of a Terwilliger algebra of a Johnson scheme depends on the characteristic of the coefficient field. We also give some other examples.
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On the Terwilliger algebras of quasi-thin Schurian association schemes
本文给出拟薄结合方案为三重传递的充要条件，并由此构造无穷多新的三重传递结合方案族。
Terwilliger algebras of generalized wreath products of association schemes
本文正式证明了对称结合方案的广义 wreath 积可适用于任意结合方案，并构建了该积结构的邻接代数与 Terwilliger 代数的不可约表示，完善了这一统一 wreath 积与直积的结构理论。
In [3], Hanaki defined the Terwilliger algebras of association schemes over a commutative unital ring. In this paper, we call the Terwilliger algebras of association schemes over a field $\mathbb{F}$ the Terwilliger $\mathbb{F}$-algebras of association schemes and study the Terwilliger $\mathbb{F}$-algebras of quasi-thin association schemes. As main results, we determine the $\mathbb{F}$-dimensions, the semisimplicity, the Jacobson radicals, and the algebraic structures of the Terwilliger $\mathbb{F}$-algebras of quasi-thin association schemes. We also get some results with independent interes…
Let $p$ denote a prime number. In this note, we focus on the modular Terwilliger algebras of association schemes defined in [3]. We define the primary module of a modular Terwilliger algebra of an association scheme and determine all its composition factors up to isomorphism. We then characterize the $p'$-valenced association schemes by some properties of their modular Terwilliger algebras. The corollaries about the modular Terwilliger algebras of association schemes are given.
On the Terwilliger algebra of the group association scheme of $C_n \rtimes C_2$
本文将 Bannai 和 Munemasa 关于阿贝尔群与二面体群 Terwilliger 代数维数的结果推广到半直积 $C_n\rtimes C_2$ 的群结合方案，并完整刻画其 Wedderburn 分量。
Terwilliger Algebra of the Ordered Hamming Scheme
本文针对有序Hamming方案的Terwilliger代数展开研究，拓展了此前关于Hamming方案及一类结合方案圈积的Terwilliger代数结构的两项结论，完善了该领域的代数结构研究。
On Terwilliger $\mathbb{F}$-algebras of factorial association schemes II
本文继续研究阶乘结合方案的Terwilliger $\mathbb{F}$-代数，得到了其所有块幂等元，并计算了块代数的$\mathbb{F}$-维数、中心和Jacobson根。
On Terwilliger $\mathbb{F}$-algebras of direct products of group divisible association schemes
本文研究群可除结合方案直积的 Terwilliger $\mathbb{F}$-代数，获得了其中心、半单性、Jacobson 根及幂零指数和 Wedderburn-Artin 分解等结构性质。
On Terwilliger $\mathbb{F}$-algebras of factorial association schemes
本文研究阶乘结合方案的 Terwilliger $\mathbb{F}$-代数，确定其中心、半单性、Jacobson 根、Wedderburn-Artin 分解，并刻画对称或 Frobenius 的情形。
On Terwilliger $\mathbb{F}$-algebras of factorial association schemes
本文研究了阶乘结合方案的Terwilliger $\mathbb{F}$-代数，确定了其维数、中心、半单性、Jacobson根及代数结构。
Terwilliger $\mathbb{F}$-algebras of certain Cayley tables
本文确定了初等阿贝尔2-群凯莱表诱导的结合方案的Terwilliger $\mathbb{F}$-代数结构，提供了无穷多例子与反例。
Almost Commutative Terwilliger Algebras I: The Group Association Scheme
本文研究由群结合方案构造的 Terwilliger 代数的几乎交换性，给出第六个等价条件，并分类了产生几乎交换 Terwilliger 代数的有限群（阿贝尔群或 Camina 群），同时计算了代数维数并给出圈积分解与原始幂等元。
Almost Commutative Terwilliger Algebras of Group Association Schemes I: Classification
本文分类了群结合方案产生几乎交换 Terwilliger 代数的群，证明其必为 Abel 群或 Camina 群，并计算前三类群对应代数的维数与非本原幂等元。
Almost Commutative Terwilliger Algebras of Group Association Schemes II: Primitive Idempotents
本文延续前期对群结合方案几乎交换 Terwilliger 代数的分类工作，针对此前未解决的第四类群，确定其 Terwilliger 代数 T(G) 的本原幂等元。
In this paper, we study the subconstituent algebras, also called as Terwilliger algebras, of association schemes that are obtained as the wreath product of one-class association schemes $K_n=H(1, n)$ for $n\ge 2$. We find that the $d$-class association scheme $K_{n_{1}}\wr K_{n_{2}} \wr ... \wr K_{n_{d}}$ formed by taking the wreath product of $K_{n_{i}}$ has the triple-regularity property. We determine the dimension of the Terwilliger algebra for the association scheme $K_{n_{1}}\wr K_{n_{2}}\wr ... \wr K_ {n_{d}}$. We give a description of the structure of the Terwilliger algebra for the wre…
Structure of Terwilliger algebras of quasi-thin association schemes
本文研究拟薄结合方案的 Terwilliger 代数，证明其在任意域上为拟遗传胞腔代数，并在特征 $2$ 下刻画基本代数结构，从而完全确定其同调与表示论性质。
On the Terwilliger algebra of the group association scheme of the symmetric group $\operatorname {sym}(7)$
本文确定了计算复杂且对寻找最大置换码有潜在应用的对称群 $\operatorname{sym}(7)$ 共轭类结合方案的 Terwilliger 代数的维数、Wedderburn 分解和块维数分解。
The Terwilliger algebras of doubly regular tournaments
本研究探讨对应双正则竞赛图的非对称秩3结合方案的Terwilliger代数性质，明确其薄不可约模与维数特征，验证了阶23内复Terwilliger代数可区分方案，而阶27时需用有理Terwilliger代数列表区分。
Let $Y$ denote a $D$-class symmetric association scheme with $D \geq 3$, and suppose $Y$ is almost-bipartite P- and Q-polynomial. Let $x$ denote a vertex of $Y$ and let $T=T(x)$ denote the corresponding Terwilliger algebra. We prove that any irreducible $T$-module $W$ is both thin and dual thin in the sense of Terwilliger. We produce two bases for $W$ and describe the action of $T$ on these bases. We prove that the isomorphism class of $W$ as a $T$-module is determined by two parameters, the dual endpoint and diameter of $W$. We find a recurrence which gives the multiplicities with which the i…
We give a full description of the algebraic structures of the Bose-Mesner algebra and Terwilliger algebra of the wreath product of one-class association schemes.
Schur rings over cyclic groups having Almost Commutative Terwilliger algebras
分类有限循环群上产生几乎交换Terwilliger代数的轨道Schur环，并给出Schur环的楔积产生几乎交换Terwilliger代数的条件。
Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand …
Let Y denote a symmetric association scheme which is Q-polynomial with respect to an ordering E_0,...,E_D of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities m_0,...,m_D is unimodal. We prove that if Y is dual-thin in the sense of Terwilliger, then the sequence of multiplicities satisfies m_i <= m_{i+1} and m_i <= m_{D-i} for i < D/2.
Almost Commutative Terwilliger Algebras II: Strong Gelfand Pairs
本文研究由有限群强 Gelfand 对经 Schur 环构造的 Terwilliger 代数，并完全分类了其中几乎交换的情形。
We prove an Assmus-Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with $s$ classes). This in particular generalizes the Assmus-Mattson-type theorems for $\mathbb{Z}_4$-linear codes due to Tanabe (2003) and Shin, Kumar, and Helleseth (2004), as well as the original theorem by Assmus and Mattson (1969). The weights of a code are $s$-tuples of non-negative integers in this case, and the conditions in our theorem for obtaining $t$-designs from the code involve concepts from polynomial interpolation in $s$ variables. The Terwillig…
Motivated by the similarities between the theory of spherical $t$-designs and that of $t$-designs in $Q$-polynomial association schemes, we study two versions of relative $t$-designs, the counterparts of Euclidean $t$-designs for $P$- and/or $Q$-polynomial association schemes. We develop the theory based on the Terwilliger algebra, which is a noncommutative associative semisimple $\mathbb{C}$-algebra associated with each vertex of an association scheme. We compute explicitly the Fisher type lower bounds on the sizes of relative $t$-designs, assuming that certain irreducible modules behave nice…
The well-known absolute bound condition for a primitive symmetric association scheme (X,S) gives an upper bound for |X| in terms of |S| and the minimal non-principal multiplicity of the scheme. In this paper we prove another upper bounds for |X| for an arbitrary primitive scheme (X,S). They do not depend on |S| but depend on some invariants of its adjacency algebra KS where K is an algebraic number field or a finite field.
We can define the adjacency algebra of an association scheme over arbitrary field. It is not always semisimple over a field of positive characteristic. The structures of adjacency algebras over a field of positive characteristic have not been sufficiently studied. In this paper, we consider the structures of adjacency algebras of dual polar schemes over a field of positive characteristic and determine them when their algebras are local algebras.
In this paper, for any prime $p$, we propose the notion of a $p$-transitive association scheme. This notion aims to generalize the fact that the regular module of a group algebra of a finite group has a unique trivial submodule to the case of the regular modules of modular adjacency algebras. We completely determine the $p$-transitive quasi-thin association schemes and the $p$-transitive association schemes with thin thin residue by their structure theory properties.
Frame Numbers and Jacobson Radicals for Partial Geometries and Related Coherent Configurations
本文研究了部分几何体和相关一致配置的模表示论，通过框架数和Jacobson根的分析，探讨了邻接代数的结构及其在不同模下的性质。
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