Multialternating graded polynomials and growth of polynomial identities
Authors:
Eli Aljadeff and Antonio Giambruno
Journal:
Proc. Amer. Math. Soc. 141 (2013), 30553065
MSC (2010):
Primary 16R50, 16P90, 16R10, 16W50
Published electronically:
June 5, 2013
MathSciNet review:
3068959
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Additional Information
Abstract: Let be a finite group and a finite dimensional graded algebra over a field of characteristic zero. When is simple as a graded algebra, by means of Regev central polynomials we construct multialternating graded polynomials of arbitrarily large degree nonvanishing on . As a consequence we compute the exponential rate of growth of the sequence of graded codimensions of an arbitrary graded algebra satisfying an ordinary polynomial identity. If , is the sequence of graded codimensions of , we prove that , the exponent of , 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 is abelian.
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 1.
 E. Aljadeff, A. Giambruno and D. La Mattina, Graded polynomial identities and exponential growth, J. Reine Angew. Math. 650 (2011), 83100. MR 2770557
 2.
 E. Aljadeff and A. KanelBelov, Representability and Specht problem for Ggraded algebras, Adv. Math. 225 (2010), 23912428. MR 2680170
 3.
 Y. A. Bahturin and V. Drensky, Graded polynomial identities of matrices, Linear Algebra Appl. 357 (2002), 1534. MR 1935223 (2003k:16034)
 4.
 Y. A. Bahturin, S. K. Sehgal and M. V. Zaicev, Finitedimensional simple graded algebras, Sb. Math. 199 (2008), no. 7, 965983. MR 2488221 (2009k:16080)
 5.
 A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. Math. 64 (1987), 118175. MR 884183 (88i:20006)
 6.
 A. Berele and A. Regev, Exponential growth for codimensions of some p.i. algebras, J. Algebra 241 (2001), 118145. MR 1838847 (2002k:16046)
 7.
 C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. MR 1013113 (90g:16001)
 8.
 E. Formanek, A conjecture of Regev about the Capelli polynomial, J. Algebra 109 (1987), 93114. MR 898339 (88i:16019)
 9.
 A. Giambruno and D. La Mattina, Graded polynomial identities and codimensions: computing the exponential growth, Adv. Math. 225 (2010), 859881. MR 2671182 (2011f:16047)
 10.
 A. Giambruno and A. Regev, Wreath products and P.I. algebras, J. Pure Applied Algebra 35 (1985), 133149. MR 775466 (86e:16027)
 11.
 A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145155. MR 1658530 (99k:16049)
 12.
 A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Adv. Math. 142 (1999), 221243. MR 1680198 (2000a:16048)
 13.
 A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Mathematical Surveys and Monographs, Vol. 122, AMS, Providence, R.I., 2005. MR 2176105 (2006g:16054)
 14.
 G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications, Vol. 16, AddisonWesley, London, 1981. MR 644144 (83k:20003)
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 A. Kemer, Ideals of Identities of Associative Algebras, AMS Translations of Mathematical Monographs, Vol. 87, Providence, R.I., 1991. MR 1108620 (92f:16031)
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 D. S. Passman, Infinite Crossed Products, Academic Press, San Diego, CA, 1989. MR 979094 (90g:16002)
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 D. Quinn, Groupgraded rings and duality, Trans. Amer. Math. Soc. 292 (1985), no. 1, 155167. MR 805958 (87d:16002)
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 E. Taft, Orthogonal conjugacies in associative and Lie algebras, Trans. Amer. Math. Soc. 113 (1964), 1829. MR 0163930 (29:1229)
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Additional Information
Eli Aljadeff
Affiliation:
Department of Mathematics, TechnionIsrael 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/S000299392013115893
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.
