发表机构
Université Paris-Saclay; University of Minnesota（巴黎萨克雷大学; 明尼苏达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用带标记非交叉弧图实现Okada代数与Okada幺半群，证明其维数、正则性等性质，构造其胞腔基并提出相关猜想，推进对称群与Okada代数的类比研究。
AI 中文摘要
众所周知，杨格（Young lattice）是对称群的布拉泰利图（Bratelli diagram），刻画了不可约表示从$\boldsymbol{\frak{S}}_{N}$到$\boldsymbol{\frak{S}}_{N-1}$的限制关系。1975年，斯坦利（Stanley）发现了一个类似的格，称为杨-斐波那契格（Young-Fibonacci lattice）；1994年，冈田（Okada）将其确定为一族代数$\boldsymbol{\big{\textbf{O}}}_N(X,Y)_{N \u2265 0}$的布拉泰利图。本文中，我们首先利用 Temperley-Lieb 代数与琼斯幺半群描述中出现的带标记非交叉弧图，实现了 Okada 代数$\boldsymbol{\textbf{O}}_N(X,Y)$及相关的 Okada 幺半群$\boldsymbol{\textbf{O}}_N$。对于一般参数$(X,Y)$，我们证明 Okada 代数$\boldsymbol{\textbf{O}}_N(X,Y)$的维数为$N!$，注意到 Okada 仅在半单情形下证明了该结果。我们将置换与带标记弧图之间的自然双射解释为与杨-斐波那契格相关的 Fomin 版 Robinson-Schensted 对应关系的体现。弧图形式使我们能够探究 Okada 幺半群与代数的结构，特别地，我们证明 Okada 幺半群是正则、非周期的$*$-幺半群，并描述了其格林关系（Green relations）与序。这些结果使我们能够构造 Okada 代数的胞腔基，并证明$\boldsymbol{\big{\textbf{O}}}_N(X,Y)_{N \u2265 0}$构成 Goodman 和 Graber 意义下的连贯胞腔代数塔。我们提出若干猜想，将每个胞腔模所附不变双线性型的 Gram 行列式用 Okada 的克隆 Schur 函数表示。最后，我们介绍两个后续的在研项目，以及一系列进一步推进对称群与 Okada 代数类比关系的问题。
英文摘要
It is well known that the Young lattice is the Bratelli diagram of the symmetric groups, expressing how irreducible representations restrict from $\mathfrak{S}_{N}$ to $\mathfrak{S}_{N-1}$. In 1975, Stanley discovered a similar lattice called the Young-Fibonacci lattice which was identified as the Bratelli diagram of a family of algebras $\{\mathbf{O}_N(X,Y)\}_{N \geq 0}$ by Okada in 1994. In this paper, we first realize the Okada algebra $\mathbf{O}_N(X,Y)$ and the associated monoid $\mathbf{O}_N$ using a labelled version of non-crossing arc-diagrams appearing in the description of the Temperley-Lieb algebra and Jones monoid. We establish, for general parameters $(X,Y)$, that the dimension of the Okada algebra $\mathbf{O}_N(X,Y)$ is $N!$, noting that Okada proved this result only in the semisimple case. We interpret a natural bijection between permutations and labelled arc-diagrams as an incarnation of Fomin's version of the Robinson-Schensted correspondence associated to the Young-Fibonacci lattice. The arc-diagram formalism allow us to probe the structure of the Okada monoid and algebra. In particular we prove that the Okada monoid is a regular, aperiodic $*$-monoid and we describe its Green relations and order. These results allow us to construct a cellular basis of the Okada algebra and to show that $\{\mathbf{O}_N(X,Y)\}_{N \geq 0}$ forms a coherent tower of cellular algebras in the sense of Goodman and Graber. We present some conjectures expressing the Gram determinant of the invariant bilinear form attached to each cell module in terms of Okada's clone Schur functions. We conclude the paper by presenting two follow-up, ongoing projects along with a series of questions pushing further the analogy between the symmetric groups and the Okada algebras.
CommentsPDFlatex, 71 pages, 27 figures
