Periodicity in bilinear lattices and the Coxeter formalism

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• 作者：Andrzej Mró ; z^a ; ^{amroz@mat.umk.pl" class="auth_mail" title="E-mail the corresponding author} ; José ; Antonio de la Peñ ; a^b ; ^{jap@cimat.mx" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A63 ; 15B36 ; 16E20 ; 16G20 ; 16G70 ; 16E35 ; 11C08
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：493
• 期：Complete
• 页码：227-260
• 全文大小：613 K

文摘

We introduce and study in detail so-called circulant (Coxeter-periodic) elements and circulant families in a bilinear lattice g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006953&_mathId=si1.gif&_user=111111111&_pii=S0024379515006953&_rdoc=1&_issn=00243795&md5=dc0e3255043d67b267680c346b3d8ae4" title="Click to view the MathML source">Kgif" overflow="scroll">K as well as their dual versions, called anti-circulant. We show that they form a natural environment for a systematic explanation of certain cyclotomic factors of the Coxeter polynomial g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006953&_mathId=si2.gif&_user=111111111&_pii=S0024379515006953&_rdoc=1&_issn=00243795&md5=4d60c74b424b1b89d35b3caa7f9f0e87" title="Click to view the MathML source">χ_Kgif" overflow="scroll">χK of g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006953&_mathId=si1.gif&_user=111111111&_pii=S0024379515006953&_rdoc=1&_issn=00243795&md5=dc0e3255043d67b267680c346b3d8ae4" title="Click to view the MathML source">Kgif" overflow="scroll">K and in consequence, of Coxeter polynomials of algebras of finite global dimension. We discuss the properties of quadratic forms induced by circulant and anti-circulant families. Moreover, we interpret the results in the language of representation theory of algebras and point out applications (facts concerning tubular families in Auslander–Reiten quivers and quadratic forms of algebras). Abstract considerations in bilinear lattices are illustrated with a collection of non-trivial examples arising from module and derived categories. The results show that techniques of linear algebra and number theory provide efficient tools for explaining various representation theoretic facts.