Abstract
Let {c n (St k )} and {c n (C k )} be the sequences of codimensions of the T-ideals generated by the standard polynomial of degreek and by thek-th Capelli polynomial, respectively. We study the asymptotic behaviour of these two sequences over a fieldF of characteristic zero. For the standard polynomial, among other results, we show that the following asymptotic equalities hold:
whereM k (F) is the algebra ofk×k matrices andM k×l (F) is the algebra of (K+l)×(k+l) matrices having the lastl rows and the lastk columns equal to zero. The precise asymptotics ofc n (M k (F)) are known and those ofM k×2k (F) andM 2k×k (F) can be easily deduced. For Capelli polynomials we show that also upper block triangular matrix algebras come into play.
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References
A. Berele and A. Regev,On the codimensions of the verbally prime P.I. algebras, Israel Journal of Mathematics91 (1995), 239–247.
A. Berele and A. Regev,Codimensions of products and of intersections of verbally prime T-ideals, Israel Journal of Mathematics103 (1998), 17–28.
A. Berele and A. Regev,Exponential growth for codimensions of some p.i. algebras, Journal of Algebra241 (2001), 118–145.
C. Curtis and I. Reiner,Representation Theory of Finite Groups and Associative Algebras, Wiley, New York, 1962.
A. Giambruno and M. Zaicev,On codimension growth of finitely generated associative algebras, Advances in Mathematics140 (1998), 145–155.
A. Giambruno and M. Zaicev,Exponential condimension growth of P.I. algebras: an exact estimate, Advances in Mathematics142 (1999), 221–243.
A. Giambruno and M. Zaicev,Minimal varieties of algebras of exponential growth, Electronic Research Announcements of the American Mathematical Society6 (2000), 40–44.
A. Giambruno and M. Zaicev,Minimal varieties of algebras of exponential growth, Advances in Mathematics (to appear).
A. Giambruno and M. Zaicev,Asymptotics of the functions of codimension growth of the standard and Capelli identities, Uspekhi Mathematicheskikh Nauk (to appear).
A. Guterman and A. Regev,On the growth of identities, inAlgebra (Moscow, 1998), de Gruyter, Berlin, 2000, pp. 319–330.
A. Kemer,Ideals of Identities of Associative Algebras, Translations of Mathematical Monographs, Vol. 87, American Mathematical Society, Providence, RI, 1988.
D. Krakowski and A. Regev,The polynomial identities of the Grassmann algebra, Transactions of the American Mathematical Society181 (1973), 429–438.
S. Mishchenko, A. Regev and M. Zaicev,The exponential growth of codimensions for Capelli identities, Israel Journal of Mathematics115 (2000), 333–342.
A. Regev,Existence of identities in A⊗B, Israel Journal of Mathematics11 (1972), 131–152.
A. Regev,Codimensions and trace codimensions of matrices are asymptotically equal, Israel Journal of Mathematics47 (1984), 246–250.
L. H. Rowen,Polynomial Identities in Ring Theory, Academic Press, New York, 1980.
E. J. Taft,Invariant Wedderburn factors, Illinois Journal of Mathematics1 (1957), 565–573.
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The first author was partially supported by MURST of Italy.
The second author was partially supported by RFBR grants 99-01-00233 and 00-15-96128.
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Giambruno, A., Zaicev, M. Asymptotics for the standard and the Capelli identities. Isr. J. Math. 135, 125–145 (2003). https://doi.org/10.1007/BF02776053
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DOI: https://doi.org/10.1007/BF02776053