特殊列代数 $$\mathfrak g_{2}$ 的基里洛夫多项式

Q3 Mathematics Arnold Mathematical Journal Pub Date : 2024-04-04 DOI:10.1007/s40598-024-00247-8

Martin T. Luu

{"title":"特殊列代数 $$\\mathfrak g_{2}$ 的基里洛夫多项式","authors":"Martin T. Luu","doi":"10.1007/s40598-024-00247-8","DOIUrl":null,"url":null,"abstract":"<div><p>As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of <i>q</i> elements and fixed Jordan type. One obtains polynomials with respect to <i>q</i> with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra <span>\$\\mathfrak g_2\$</span>. In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-024-00247-8.pdf","citationCount":"0","resultStr":"{\"title\":\"Kirillov Polynomials for the Exceptional Lie Algebra \\\$\\\\mathfrak g_{2}\\\$\",\"authors\":\"Martin T. Luu\",\"doi\":\"10.1007/s40598-024-00247-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of <i>q</i> elements and fixed Jordan type. One obtains polynomials with respect to <i>q</i> with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra <span>\\\$\\\\mathfrak g_2\\\$</span>. In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.</p></div>\",\"PeriodicalId\":37546,\"journal\":{\"name\":\"Arnold Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-04-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s40598-024-00247-8.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Arnold Mathematical Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40598-024-00247-8\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Arnold Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40598-024-00247-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}

引用次数: 0

摘要

作为轨道方法发展的一部分，基里洛夫计算了在具有 q 个元素和固定约旦类型的有限域中具有系数的严格上三角矩阵的数量。我们得到了关于 q 的多项式，它们具有许多有趣的性质，并与 A 型表示理论密切相关。在本研究中，我们发展了例外李代数（\mathfrak g_2\）的相应理论。我们特别指出，前导系数可以用 Springer 对应关系来表示。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

Kirillov Polynomials for the Exceptional Lie Algebra $\mathfrak g_{2}$

As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of q elements and fixed Jordan type. One obtains polynomials with respect to q with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra $\mathfrak g_2$. In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Arnold Mathematical Journal Mathematics-Mathematics (all)

CiteScore

1.50

自引率

0.00%

发文量

期刊介绍： The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.

期刊最新文献

Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces Kirillov Polynomials for the Exceptional Lie Algebra $\mathfrak g_{2}$ Polynomial Superpotential for Grassmannian ${\text {Gr}}(k,n)$ from a Limit of Vertex Function A Note on Contact Manifolds with Infinite Fillings Higher Dimensional Versions of Theorems of Euler and Fuss