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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Polynomial detection of matrix subalgebras


Author: Daniel Birmajer
Journal: Proc. Amer. Math. Soc. 133 (2005), 1007-1012
MSC (2000): Primary 15A24, 15A99, 16R99
Published electronically: October 18, 2004
MathSciNet review: 2117201
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Abstract: The double Capelli polynomial of total degree $2t$ is

\begin{displaymath}\sum \left\{ (\mathrm{sg}\, \sigma\tau) x_{\sigma(1)}y_{\tau(... ...\sigma(t)}y_{\tau(t)} \vert\; \sigma,\, \tau \in S_t\right\}. \end{displaymath}

It was proved by Giambruno-Sehgal and Chang that the double Capelli polynomial of total degree $4n$ is a polynomial identity for $M_n(F)$. (Here, $F$ is a field and $M_n(F)$ is the algebra of $n \times n$ matrices over $F$.) Using a strengthened version of this result obtained by Domokos, we show that the double Capelli polynomial of total degree $4n-2$ is a polynomial identity for any proper $F$-subalgebra of $M_n(F)$. Subsequently, we present a similar result for nonsplit inequivalent extensions of full matrix algebras.


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Additional Information

Daniel Birmajer
Affiliation: Department of Mathematics and Computer Science, Nazareth College, 4245 East Avenue, Rochester, New York 14618
Email: abirmaj6@naz.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07631-2
PII: S 0002-9939(04)07631-2
Keywords: Polynomial identity, polynomial test, matrix subalgebra, double Capelli polynomial
Received by editor(s): November 13, 2003
Received by editor(s) in revised form: December 22, 2003
Published electronically: October 18, 2004
Communicated by: Martin Lorenz
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.