Graded polynomial identities for matrices with the transpose involution

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• 作者：Darrell Haile^a ; ^{haile@indiana.edu" class="auth_mail" title="E-mail the corresponding author} ; Michael Natapov^b ; ^{natapov@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16S35 ; 16R10 ; 16W10 ; 16W50 ; 05C30 ; 05C45
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 October 2016
• 年：2016
• 卷：464
• 期：Complete
• 页码：175-197
• 全文大小：510 K

文摘

Let G be a group of order k  . We consider the algebra M_k(C)$M_k(\mathbb{C})$ of k by k matrices over the complex numbers and view it as a crossed product with respect to G by embedding G   in the symmetric group S_k$S_k$ via the regular representation and embedding S_k$S_k$ in M_k(C)$M_k(\mathbb{C})$ in the usual way. This induces a natural G  -grading on M_k(C)$M_k(\mathbb{C})$ which we call a crossed-product grading. We study the graded ⁎-identities for M_k(C)$M_k(\mathbb{C})$ equipped with such a crossed-product grading and the transpose involution. To each multilinear monomial in the free graded algebra with involution we associate a directed labeled graph. Use of these graphs allows us to produce a set of generators for the (T,⁎)$(T,⁎)$-ideal of identities. It also leads to new proofs of the results of Kostant and Rowen on the standard identities satisfied by skew matrices. Finally we determine an asymptotic formula for the ⁎-graded codimension of M_k(C)$M_k(\mathbb{C})$.