Asymptotics for the multiplicities in the cocharacters of some PIalgebras
Authors:
Francesca Benanti, Antonio Giambruno and Irina Sviridova
Journal:
Proc. Amer. Math. Soc. 132 (2004), 669679
MSC (2000):
Primary 16R10, 16P90
Published electronically:
August 13, 2003
MathSciNet review:
2019941
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider associative PIalgebras 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.
 1.
A.
Berele and A.
Regev, Applications of hook Young diagrams to P.I. algebras,
J. Algebra 82 (1983), no. 2, 559–567. MR 704771
(84g:16012), http://dx.doi.org/10.1016/00218693(83)901679
 2.
Allan
Berele and Amitai
Regev, On the codimensions of the verbally prime P.I.
algebras, Israel J. Math. 91 (1995), no. 13,
239–247. MR 1348314
(96g:16028), http://dx.doi.org/10.1007/BF02761648
 3.
A.
Berele and A.
Regev, Codimensions of products and of intersections of verbally
prime 𝑇ideals, Israel J. Math. 103 (1998),
17–28. MR
1613536 (99b:16037), http://dx.doi.org/10.1007/BF02762265
 4.
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, New YorkLondon, 1962. MR 0144979
(26 #2519)
 5.
Vesselin
Drensky, Free algebras and PIalgebras, SpringerVerlag
Singapore, Singapore, 2000. Graduate course in algebra. MR 1712064
(2000j:16002)
 6.
Vesselin
Drensky, Codimensions of 𝑇ideals and Hilbert series of
relatively free algebras, J. Algebra 91 (1984),
no. 1, 1–17. MR 765766
(86b:16010), http://dx.doi.org/10.1016/00218693(84)901212
 7.
Veselin
S. Drenski, Extremal varieties of algebras. I, Serdica
13 (1987), no. 4, 320–332 (Russian). MR 929452
(90e:08010)
Veselin
S. Drenski, Extremal varieties of algebras. II, Serdica
14 (1988), no. 1, 20–27 (Russian). MR 944480
(90e:08011)
 8.
V. S. Drensky and M. Kassabov, Growth of nonmatrix varieties of algebras, preprint.
 9.
Edward
Formanek, Invariants and the ring of generic matrices, J.
Algebra 89 (1984), no. 1, 178–223. MR 748233
(85g:15031), http://dx.doi.org/10.1016/00218693(84)902400
 10.
A.
Giambruno and M.
Zaicev, On codimension growth of finitely generated associative
algebras, Adv. Math. 140 (1998), no. 2,
145–155. MR 1658530
(99k:16049), http://dx.doi.org/10.1006/aima.1998.1766
 11.
A.
Giambruno and M.
Zaicev, Exponential codimension growth of PI algebras: an exact
estimate, Adv. Math. 142 (1999), no. 2,
221–243. MR 1680198
(2000a:16048), http://dx.doi.org/10.1006/aima.1998.1790
 12.
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
(2001i:16041)
 13.
A.
Giambruno and M.
Zaicev, Minimal varieties of algebras of
exponential growth, Electron. Res. Announc.
Amer. Math. Soc. 6
(2000), 40–44 (electronic). MR 1767635
(2001e:16037), http://dx.doi.org/10.1090/S1079676200000780
 14.
A. Giambruno and M. Zaicev, Minimal varieties of algebras of exponential growth, Adv. Math. 174 (2003), 3342.
 15.
Gordon
James and Adalbert
Kerber, The representation theory of the symmetric group,
Encyclopedia of Mathematics and its Applications, vol. 16,
AddisonWesley Publishing Co., Reading, Mass., 1981. With a foreword by P.
M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
(83k:20003)
 16.
A.
R. Kemer, The Spechtian nature of 𝑇ideals whose
condimensions have power growth, Sibirsk. Mat. Ž.
19 (1978), no. 1, 54–69, 237 (Russian). MR 0466190
(57 #6070)
 17.
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
(92f:16031)
 18.
D.
Krakowski and A.
Regev, The polynomial identities of the
Grassmann algebra, Trans. Amer. Math. Soc.
181 (1973),
429–438. MR 0325658
(48 #4005), http://dx.doi.org/10.1090/S00029947197303256585
 19.
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 (58 #27695)
 20.
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
(46 #3561)
 21.
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
(2000e:16026), http://dx.doi.org/10.1006/jabr.1998.7916
 22.
A.
P. Popov, Identities of the tensor square of a Grassmann
algebra, Algebra i Logika 21 (1982), no. 4,
442–471 (Russian). MR 721348
(85j:15018)
 23.
C.
Procesi, Computing with 2×2 matrices, J. Algebra
87 (1984), no. 2, 342–359. MR 739938
(86g:16022), http://dx.doi.org/10.1016/00218693(84)901418
 24.
Amitai
Regev, Existence of identities in 𝐴⊗𝐵,
Israel J. Math. 11 (1972), 131–152. MR 0314893
(47 #3442)
 25.
Amitai
Regev, Codimensions and trace codimensions of matrices are
asymptotically equal, Israel J. Math. 47 (1984),
no. 23, 246–250. MR 738172
(85j:16024), http://dx.doi.org/10.1007/BF02760520
 26.
A.
N. StoyanovaVenkova, 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 (84b:08007)
 1.
 A. Berele and A. Regev, Applications of hook Young diagrams to P.I. algebras, J. Algebra 82 (1983), 559567. MR 84g:16012
 2.
 A. Berele and A. Regev, On the codimensions of the verbally prime P.I. algebras, Israel J. Math. 91 (1995), 239247. MR 96g:16028
 3.
 A. Berele and A. Regev, Codimensions of products and of intersections of verbally prime Tideals, Israel J. Math. 103 (1998), 1728. MR 99b:16037
 4.
 C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, John Wiley and Sons, New York, 1962. MR 26:2519
 5.
 V. Drensky, Free Algebras and PIAlgebras, SpringerVerlag, BerlinHeidelbergSingapore, 1999. MR 2000j:16002
 6.
 V. S. Drensky, Codimensions of Tideals and Hilbert series of relatively free algebras, J. Algebra 91 (1984), 117.MR 86b:16010
 7.
 V. S. Drensky, Extremal varieties of algebras. I, II, Serdica 13 (1987), 320332; 14 (1988), 2027 (Russian). MR 90e:08010, MR 90e:08011
 8.
 V. S. Drensky and M. Kassabov, Growth of nonmatrix varieties of algebras, preprint.
 9.
 E. Formanek, Invariants and the ring of generic matrices, J. Algebra 89 (1984), 178223. MR 85g:15031
 10.
 A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145155. MR 99k:16049
 11.
 A. Giambruno and M. Zaicev, Exponential codimension growth of P.I. algebras: an exact estimate, Adv. Math. 142 (1999), 221243. MR 2000a:16048
 12.
 A. Giambruno and M. Zaicev, A characterization of varieties of associative algebras of exponent two, Serdica Math. J. 26 (2000), 245252. MR 2001i:16041
 13.
 A. Giambruno and M. Zaicev, Minimal varieties of algebras of exponential growth, Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 4044 (electronic). MR 2001e:16037
 14.
 A. Giambruno and M. Zaicev, Minimal varieties of algebras of exponential growth, Adv. Math. 174 (2003), 3342.
 15.
 G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, Vol. 16, AddisonWesley, London, 1981. MR 83k:20003
 16.
 A. R. Kemer, The Spechtian nature of ideals whose codimensions have power growth, Sibirskii Matematicheskii 19 (1978), 5469 (Russian). Translation: Siberian Math. J. 19 (1978), 3748. MR 57:6070
 17.
 A. R. Kemer, Ideals of Identities of Associative Algebras, Translations of Math. Monographs 87, Amer. Math. Soc., Providence, RI, 1991. MR 92f:16031
 18.
 D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973), 429438. MR 48:4005
 19.
 V. N. Latyshev, The complexity of nonmatrix varieties of associative algebras. I, II, Algebra i Logika 16 (1977), 149183, 184199 (Russian). Translation: Algebra and Logic 16 (1977), 48122, 122133. MR 58:27695
 20.
 Yu. N. Mal'cev, A basis for the identities of the algebra of upper triangular matrices, Algebra i Logika 10 (1971), 393400 (Russian). Translation: Algebra and Logic 10 (1971), 242247. MR 46:3561
 21.
 S. P. Mishchenko, A. Regev and M. V. Zaicev, A characterization of PIalgebras with bounded multiplicities of the cocharacters, J. Algebra 219 (1999), 356368.MR 2000e:16026
 22.
 A. P. Popov, Identities of the tensor square of a Grassmann algebra, Algebra i Logika 21 (1982), 442471 (Russian). Translation: Algebra and Logic 21 (1982), 296316.MR 85j:15018
 23.
 C. Procesi, Computing with matrices, J. Algebra 87 (1984), 342359.MR 86g:16022
 24.
 A. Regev, Existence of identities in , Israel J. Math. 11 (1972), 131152.MR 47:3442
 25.
 A. Regev, Codimensions and trace codimensions of matrices are asymtotically equal, Israel J. Math. 47 (1984), 246250.MR 85j:16024
 26.
 A. N. StoyanovaVenkova, Some lattices of varieties of associative algebras defined by identities of the fifth degree, C. R. Acad. Bulgare Sci. 35 (1982), 865868.MR 84b:08007
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
16R10,
16P90
Retrieve articles in all journals
with MSC (2000):
16R10,
16P90
Additional Information
Francesca Benanti
Affiliation:
Dipartimento di Matematica ed Applicazioni, Università di Palermo, via Archirafi 34, 90123 Palermo, Italy
Email:
fbenanti@math.unipa.it
Antonio Giambruno
Affiliation:
Dipartimento di Matematica ed Applicazioni, Università di Palermo, via Archirafi 34, 90123 Palermo, Italy
Email:
a.giambruno@unipa.it
Irina Sviridova
Affiliation:
Department of Algebra and Geometric Computations, Faculty of Mathematics and Mechanics, Ulyanovsk State University, Ulyanovsk 4327000, Russia
Email:
sviridova_i@rambler.ru
DOI:
http://dx.doi.org/10.1090/S000299390307093X
PII:
S 00029939(03)07093X
Keywords:
Polynomial identities,
multiplicities,
codimensions,
growth
Received by editor(s):
March 22, 2002
Received by editor(s) in revised form:
July 31, 2002, and October 30, 2002
Published electronically:
August 13, 2003
Additional Notes:
The first and the second authors were partially supported by MURST of Italy
The third author was partially supported by the scientific program “The Universities of Russia"
Communicated by:
Martin Lorenz
Article copyright:
© Copyright 2003 American Mathematical Society
