Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Asymptotic behaviour of codimensions of p. i. algebras satisfying Capelli identities


Authors: Allan Berele and Amitai Regev
Journal: Trans. Amer. Math. Soc. 360 (2008), 5155-5172
MSC (2000): Primary 16R10
DOI: https://doi.org/10.1090/S0002-9947-08-04500-5
Published electronically: May 27, 2008
MathSciNet review: 2415069
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A$ be a p. i. algebra with 1 in characteristic zero, satisfying a Capelli identity. Then the cocharacter sequence $ c_n(A)$ is asymptotic to a function of the form $ an^g\ell^n$, where $ \ell \in \mathbb{N}$ and $ g \in \mathbb{Z}$.


References [Enhancements On Off] (What's this?)

  • 1. W. Beckner and A. Regev, Asymptotics and algebraicity of some generating functions, Adv. Math. 65 (1987), 1-15. MR 893467 (88h:05008)
  • 2. W. Beckner and A. Regev, Asymptotic estimates using probability, Adv. Math. 138 (1998), 1-14. MR 1645060 (99j:05011)
  • 3. A. Ya. Belov, Rationality of Hilbert series with respect to free algebras, Russian Math. Surveys 52 (1997), 394-395. MR 1480146 (98f:16018)
  • 4. A. Berele, Approximate multiplicities in the trace cocharacter sequence of two three-by-three matrices,Comm. Algebra 25 (1997) 1975-1983. MR 1446144 (98j:05129)
  • 5. A. Berele, Applications of Belov's theorem to the cocharacter sequence of p. i. algebras, J. of Alg. 298 (2006), 208-214. MR 2215124 (2006m:16023)
  • 6. A. Berele and A. Regev, Codimensions of products and of intersections of verbally prime $ T$-ideals, Israel J. Math. 103 (1998), 17-28. MR 1613536 (99b:16037)
  • 7. A. Berele and A. Regev, Exponential growth for codimensions of some p. i. algebras, J. of Alg. 241 (2001), 118-145. MR 1838847 (2002k:16046)
  • 8. V. Drensky, Codimensions of $ T$-ideals and Hilbert series of relatively free algebras, J. of Alg. 89 (1984), 178-223. MR 765766 (86b:16010)
  • 9. V. Drensky and G. K. Genov, Multiplicities of Schur functions in invariants of two $ 3\times3$ matrices, J. Algebra 264 (2003), 496-519. MR 1981418 (2004c:16037)
  • 10. A. Giambruno and M. V. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145-155. MR 1658530 (99k:16049)
  • 11. A. Giambruno and M. V. Zaicev, Exponential codimension growth of PI algebras: An exact estimate, Adv. Math. 142 (1999), 221-243. MR 1680198 (2000a:16048)
  • 12. B. Grünbaum, Convex Polytopes, Interscience, London, 1967. MR 0226496 (37:2085)
  • 13. A. Guterman and A. Regev, On the growth of identities, In: Algebra (Moscow: 1998), 319-330, de Gruyter, Berlin, 2000. MR 1754678 (2001a:16039)
  • 14. A. Regev, Asymptotic values for degrees associated with strips of Young diagrams, Adv. in Math. 41 (1981), 115-136. MR 625890 (82h:20015)
  • 15. A. Regev, Algebras satisfying a Capelli identity, Israel J. Math. 33 (1979), 149-154. MR 571250 (81i:16022)
  • 16. G. M. Zeigler, Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995. MR 1311028 (96a:52011)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 16R10

Retrieve articles in all journals with MSC (2000): 16R10


Additional Information

Allan Berele
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
Email: aberele@condor.depaul.edu

Amitai Regev
Affiliation: Department of Theoretical Mathematics, Weizmann Institute, Rehovot, Israel
Email: amitai.regev@wisdom.weizmann.ac.il

DOI: https://doi.org/10.1090/S0002-9947-08-04500-5
Keywords: Polynomial identities, cocharacter sequence
Received by editor(s): June 5, 2006
Published electronically: May 27, 2008
Additional Notes: The work of the first author was supported by both the Faculty Research Council of DePaul University and the National Security Agency, under Grant MDA904-500270. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.
The work of the second author was partially supported by ISF grant 947-04.
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society