具有几乎交换Terwilliger代数的循环群上的Schur环
Schur rings over cyclic groups having Almost Commutative Terwilliger algebras
AI总结：
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。
AI中文摘要：
Terwilliger代数是基于结合方案构造的矩阵代数的子代数。Rie Tanaka定义了Terwilliger代数几乎交换的含义，并在结合方案可交换的情况下给出了五个等价条件。后来，从可交换Schur环出发的Terwilliger代数几乎交换的第六个条件被发现。本文首先对有限循环群上产生几乎交换Terwilliger代数的轨道Schur环进行分类。特别地，我们表明用于形成轨道Schur环的自同构子群要么是平凡的，要么当循环群具有素数幂阶时是整个自同构群。如果循环群阶为$2^n$，这些是仅有的选项。如果循环群阶为$p^n$（$p$为奇素数），则阶为$p^{n-1}$的自同构子群也适用。如果群具有非素数幂阶，则唯一产生几乎交换Terwilliger代数的轨道Schur环来自自同构群的平凡子群。然后我们给出Schur环的楔积产生几乎交换Terwilliger代数的条件。这使我们能够精确确定循环群上的Schur环何时产生几乎交换Terwilliger代数。
英文摘要：
Terwilliger algebras are subalgebras of a matrix algebra constructed from an association scheme. Rie Tanaka defined what it means for a Terwilliger algebra to be almost commutative and gave five equivalent conditions in the case where the association scheme is commutative. A sixth condition for a Terwilliger algebra coming from a commutative Schur ring to be almost commutative has since been discovered. In this paper we first provide a classification of orbit Schur rings that produce an almost commutative Terwilliger algebra for a finite cyclic group. In particular, we show that the subgroup of automorphisms used to form the orbit Schur ring is either trivial, or the whole automorphism group when the cyclic group has prime power order. If the cyclic group has order $2^n$, these are the only options. If the cyclic group has order $p^n$, for an odd prime $p$, then the automorphism subgroup of order $p^{n-1}$ also works. If the group has a non-prime power order then the only orbit Schur ring that produces an almost commutative Terwilliger algebra comes from the trivial subgroup of the automorphism group. We then give a condition for when a wedge product of Schur rings produces an almost commutative Terwilliger algebra. This allows us to determine exactly when a Schur ring over a cyclic group produces an almost commutative Terwilliger algebra.
