{"title":"Matrix Spherical Functions on Finite Groups","authors":"C. Blanco Villacorta, I. Pacharoni, J. A. Tirao","doi":"10.1007/s00041-024-10110-1","DOIUrl":null,"url":null,"abstract":"<p>In this paper we focus our attention on matrix or operator-valued spherical functions associated to finite groups (<i>G</i>, <i>K</i>), where <i>K</i> is a subgroup of <i>G</i>. We introduce the notion of matrix-valued spherical functions on <i>G</i> associated to any <i>K</i>-type <span>\\(\\delta \\in \\hat{K}\\)</span> by means of solutions of certain associated integral equations. The main properties of spherical functions are established from their characterization as eigenfunctions of right convolution multiplication by functions in <span>\\(A[G]^K\\)</span>, the algebra of <i>K</i>-central functions in the group algebra <i>A</i>[<i>G</i>]. The irreducible representations of <span>\\(A[G]^K\\)</span> are closely related to the irreducible spherical functions on <i>G</i>. This allows us to study and compute spherical functions via the representations of this algebra.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00041-024-10110-1","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we focus our attention on matrix or operator-valued spherical functions associated to finite groups (G, K), where K is a subgroup of G. We introduce the notion of matrix-valued spherical functions on G associated to any K-type \(\delta \in \hat{K}\) by means of solutions of certain associated integral equations. The main properties of spherical functions are established from their characterization as eigenfunctions of right convolution multiplication by functions in \(A[G]^K\), the algebra of K-central functions in the group algebra A[G]. The irreducible representations of \(A[G]^K\) are closely related to the irreducible spherical functions on G. This allows us to study and compute spherical functions via the representations of this algebra.
在本文中，我们将注意力集中在与有限群（G，K）相关联的矩阵或算子值球函数上，其中 K 是 G 的一个子群。我们通过某些相关积分方程的解引入了与任意 K 型 \(\delta \in \hat{K}\) 相关联的 G 上矩阵值球函数的概念。球函数的主要性质是从它们作为右卷积乘以群代数 A[G] 中的 K 中心函数代数 \(A[G]^K\)中函数的特征函数的特性中建立起来的。\(A[G]^K\) 的不可还原表示与 G 上的不可还原球函数密切相关。
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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