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

MathSciNet review:
3068959

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Abstract | References | Similar Articles | 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 non-vanishing 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.

**1.**Eli Aljadeff, Antonio Giambruno, and Daniela La Mattina,*Graded polynomial identities and exponential growth*, J. Reine Angew. Math.**650**(2011), 83–100. MR**2770557**, 10.1515/CRELLE.2011.004**2.**Eli Aljadeff and Alexei Kanel-Belov,*Representability and Specht problem for 𝐺-graded algebras*, Adv. Math.**225**(2010), no. 5, 2391–2428. MR**2680170**, 10.1016/j.aim.2010.04.025**3.**Yuri Bahturin and Vesselin Drensky,*Graded polynomial identities of matrices*, Linear Algebra Appl.**357**(2002), 15–34. MR**1935223**, 10.1016/S0024-3795(02)00356-7**4.**Yu. A. Bakhturin, M. V. Zaĭtsev, and S. K. Segal,*Finite-dimensional simple graded algebras*, Mat. Sb.**199**(2008), no. 7, 21–40 (Russian, with Russian summary); English transl., Sb. Math.**199**(2008), no. 7-8, 965–983 (2008). MR**2488221**, 10.1070/SM2008v199n07ABEH003949**5.**A. Berele and A. Regev,*Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras*, Adv. in Math.**64**(1987), no. 2, 118–175. MR**884183**, 10.1016/0001-8708(87)90007-7**6.**Allan Berele and Amitai Regev,*Exponential growth for codimensions of some p.i. algebras*, J. Algebra**241**(2001), no. 1, 118–145. MR**1838847**, 10.1006/jabr.2000.8672**7.**Charles W. Curtis and Irving Reiner,*Representation theory of finite groups and associative algebras*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Reprint of the 1962 original; A Wiley-Interscience Publication. MR**1013113****8.**Edward Formanek,*A conjecture of Regev about the Capelli polynomial*, J. Algebra**109**(1987), no. 1, 93–114. MR**898339**, 10.1016/0021-8693(87)90166-9**9.**A. Giambruno and D. La Mattina,*Graded polynomial identities and codimensions: computing the exponential growth*, Adv. Math.**225**(2010), no. 2, 859–881. MR**2671182**, 10.1016/j.aim.2010.03.013**10.**Antonino Giambruno and Amitai Regev,*Wreath products and P.I. algebras*, J. Pure Appl. Algebra**35**(1985), no. 2, 133–149. MR**775466**, 10.1016/0022-4049(85)90036-2**11.**A. Giambruno and M. Zaicev,*On codimension growth of finitely generated associative algebras*, Adv. Math.**140**(1998), no. 2, 145–155. MR**1658530**, 10.1006/aima.1998.1766**12.**A. Giambruno and M. Zaicev,*Exponential codimension growth of PI algebras: an exact estimate*, Adv. Math.**142**(1999), no. 2, 221–243. MR**1680198**, 10.1006/aima.1998.1790**13.**Antonio Giambruno and Mikhail Zaicev,*Polynomial identities and asymptotic methods*, Mathematical Surveys and Monographs, vol. 122, American Mathematical Society, Providence, RI, 2005. MR**2176105****14.**Gordon James and Adalbert Kerber,*The representation theory of the symmetric group*, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR**644144****15.**Aleksandr Robertovich Kemer,*Ideals of identities of associative algebras*, Translations of Mathematical Monographs, vol. 87, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by C. W. Kohls. MR**1108620****16.**Donald S. Passman,*Infinite crossed products*, Pure and Applied Mathematics, vol. 135, Academic Press, Inc., Boston, MA, 1989. MR**979094****17.**Declan Quinn,*Group-graded rings and duality*, Trans. Amer. Math. Soc.**292**(1985), no. 1, 155–167. MR**805958**, 10.1090/S0002-9947-1985-0805958-0**18.**Amitai Regev,*Existence of identities in 𝐴⊗𝐵*, Israel J. Math.**11**(1972), 131–152. MR**0314893****19.**Amitai Regev,*Asymptotic values for degrees associated with strips of Young diagrams*, Adv. in Math.**41**(1981), no. 2, 115–136. MR**625890**, 10.1016/0001-8708(81)90012-8**20.**Irina Sviridova,*Identities of pi-algebras graded by a finite abelian group*, Comm. Algebra**39**(2011), no. 9, 3462–3490. MR**2845581**, 10.1080/00927872.2011.593417**21.**Earl J. Taft,*Orthogonal conjugacies in associative and Lie algebras*, Trans. Amer. Math. Soc.**113**(1964), 18–29. MR**0163930**, 10.1090/S0002-9947-1964-0163930-7

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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

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.