Asymptotics for the multiplicities in the cocharacters of some PI-algebras

Benanti, Francesca; Giambruno, Antonio; Sviridova, Irina

doi:10.1090/S0002-9939-03-07093-X

Asymptotics for the multiplicities in the cocharacters of some PI-algebras
HTML articles powered by AMS MathViewer

by Francesca Benanti, Antonio Giambruno and Irina Sviridova

Proc. Amer. Math. Soc. 132 (2004), 669-679

DOI: https://doi.org/10.1090/S0002-9939-03-07093-X

Published electronically: August 13, 2003

PDF | Request permission

Abstract:

We consider associative PI-algebras over a field of characteristic zero. We study the asymptotic behavior of the sequence of multiplicities of the cocharacters for some significant classes of algebras. We also give a characterization of finitely generated algebras for which this behavior is linear or quadratic.

References

A. Berele and A. Regev, Applications of hook Young diagrams to P.I. algebras, J. Algebra 82 (1983), no. 2, 559–567. MR 704771, DOI 10.1016/0021-8693(83)90167-9
Allan Berele and Amitai Regev, On the codimensions of the verbally prime P.I. algebras, Israel J. Math. 91 (1995), no. 1-3, 239–247. MR 1348314, DOI 10.1007/BF02761648
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, DOI 10.1007/BF02762265
Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
Vesselin Drensky, Free algebras and PI-algebras, Springer-Verlag Singapore, Singapore, 2000. Graduate course in algebra. MR 1712064
Vesselin Drensky, Codimensions of $T$-ideals and Hilbert series of relatively free algebras, J. Algebra 91 (1984), no. 1, 1–17. MR 765766, DOI 10.1016/0021-8693(84)90121-2
John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
V. S. Drensky and M. Kassabov, Growth of nonmatrix varieties of algebras, preprint.
Edward Formanek, Invariants and the ring of generic matrices, J. Algebra 89 (1984), no. 1, 178–223. MR 748233, DOI 10.1016/0021-8693(84)90240-0
A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), no. 2, 145–155. MR 1658530, DOI 10.1006/aima.1998.1766
A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Adv. Math. 142 (1999), no. 2, 221–243. MR 1680198, DOI 10.1006/aima.1998.1790
A. Giambruno and M. Zaicev, A characterization of varieties of associative algebras of exponent two, Serdica Math. J. 26 (2000), no. 3, 245–252. MR 1803836
A. Giambruno and M. Zaicev, Minimal varieties of algebras of exponential growth, Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 40–44. MR 1767635, DOI 10.1090/S1079-6762-00-00078-0
A. Giambruno and M. Zaicev, Minimal varieties of algebras of exponential growth, Adv. Math. 174 (2003), 33–42.
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
A. R. Kemer, The Spechtian nature of $T$-ideals whose condimensions have power growth, Sibirsk. Mat. Ž. 19 (1978), no. 1, 54–69, 237 (Russian). MR 0466190
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, DOI 10.1090/mmono/087
D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973), 429–438. MR 325658, DOI 10.1090/S0002-9947-1973-0325658-5
V. N. Latyšev, The complexity of nonmatrix varieties of associative algebras. I, II, Algebra i Logika 16 (1977), no. 2, 149–183, 184–199, 249–250 (Russian). MR 0552771
Ju. N. Mal′cev, A basis for the identities of the algebra of upper triangular matrices, Algebra i Logika 10 (1971), 393–400 (Russian). MR 0304426
S. P. Mishchenko, A. Regev, and M. V. Zaicev, A characterization of P.I. algebras with bounded multiplicities of the cocharacters, J. Algebra 219 (1999), no. 1, 356–368. MR 1707676, DOI 10.1006/jabr.1998.7916
A. P. Popov, Identities of the tensor square of a Grassmann algebra, Algebra i Logika 21 (1982), no. 4, 442–471 (Russian). MR 721348
C. Procesi, Computing with $2\times 2$ matrices, J. Algebra 87 (1984), no. 2, 342–359. MR 739938, DOI 10.1016/0021-8693(84)90141-8
Amitai Regev, Existence of identities in $A\otimes B$, Israel J. Math. 11 (1972), 131–152. MR 314893, DOI 10.1007/BF02762615
Amitai Regev, Codimensions and trace codimensions of matrices are asymptotically equal, Israel J. Math. 47 (1984), no. 2-3, 246–250. MR 738172, DOI 10.1007/BF02760520
A. N. Stoyanova-Venkova, Some lattices of varieties of associative algebras defined by identities of the fifth degree, C. R. Acad. Bulgare Sci. 35 (1982), no. 7, 867–868 (Russian). MR 681740