Decomposing modular tensor products, and periodicity of ‘Jordan partitions’

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• 作者：S.P. Glasby^a ; ¹ ; ^{GlasbyS@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; ; Cheryl E. Praeger^a ; ² ; ^{Cheryl.Praeger@uwa.edu.au" class="auth_mail" title="E-mail the corresponding author} ; ; Binzhou Xia^b ; ^{binzhouxia@pku.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A69 ; 15A21 ; 13C05
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：450
• 期：Complete
• 页码：570-587
• 全文大小：462 K

文摘

Let J_r$J_r$ denote an r×r$r\times r$ matrix with minimal and characteristic polynomials (t−1)^r${(t-1)}^r$. Suppose r⩽s$r⩽s$. It is not hard to show that the Jordan canonical form of J_r⊗J_s$J_r\otimes J_s$ is similar to J_λ1⊕⋯⊕J_λr$J_{{\lambda }_1}\oplus \cdots \oplus J_{{\lambda }_r}$ where λ₁⩾⋯⩾λ_r>0${\lambda }_1⩾\cdots ⩾{\lambda }_r>0$ and ${\sum }_{i=1}^r{\lambda }_i=rs$. The partition λ(r,s,p):=(λ₁,…,λ_r)$\lambda (r,s,p):=({\lambda }_1,\dots ,{\lambda }_r)$ of rs  , which depends only on r,s$r,s$ and the characteristic p:=char(F)$p:=\mathrm{char}(F)$, has many applications including the study of algebraic groups. We prove new periodicity and duality results for λ(r,s,p)$\lambda (r,s,p)$ that depend on the smallest p-power exceeding r. This generalizes results of J.A. Green, B. Srinivasan, and others which depend on the smallest p-power exceeding the (potentially large) integer s. It also implies that for fixed r   we can construct a finite table allowing the computation of λ(r,s,p)$\lambda (r,s,p)$ for all s and p  , with s⩾r$s⩾r$ and p prime.