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

 

Multialternating graded polynomials and growth of polynomial identities


Authors: Eli Aljadeff and Antonio Giambruno
Journal: Proc. Amer. Math. Soc. 141 (2013), 3055-3065
MSC (2010): Primary 16R50, 16P90, 16R10, 16W50
Published electronically: June 5, 2013
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Abstract: Let $ G$ be a finite group and $ A$ a finite dimensional $ G$-graded algebra over a field of characteristic zero. When $ A$ is simple as a $ G$-graded algebra, by means of Regev central polynomials we construct multialternating graded polynomials of arbitrarily large degree non-vanishing on $ A$. As a consequence we compute the exponential rate of growth of the sequence of graded codimensions of an arbitrary $ G$-graded algebra satisfying an ordinary polynomial identity. If $ c_n^G(A), n=1,2,\ldots $, is the sequence of graded codimensions of $ A$, we prove that $ exp^G(A)=\lim _{n\to \infty }\sqrt [n]{c_n^G(A)}$, the $ G$-exponent of $ A$, exists and is an integer. This result was proved by the authors and D. La Mattina in 2011 and by the second author and D. La Mattina in 2010 in the case $ G$ is abelian.


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Additional Information

Eli Aljadeff
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
Email: aljadeff@tx.technion.ac.il

Antonio Giambruno
Affiliation: Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy
Email: antonio.giambruno@unipa.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11589-3
PII: S 0002-9939(2013)11589-3
Keywords: Graded algebra, polynomial identity, growth, codimensions
Received by editor(s): July 22, 2011
Received by editor(s) in revised form: December 5, 2011
Published electronically: June 5, 2013
Additional Notes: The first author was supported by the Israel Science Foundation (grant No. 1283/08) and by the E. Schaver Research Fund
The second author was partially supported by MIUR of Italy
Communicated by: Harm Derksen
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.