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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ A$-identities for the Grassmann algebra: The conjecture of Henke and Regev


Authors: Dimas José Gonçalves and Plamen Koshlukov
Journal: Proc. Amer. Math. Soc. 136 (2008), 2711-2717
MSC (2000): Primary 16R10; Secondary 16R40, 16R50
Published electronically: April 3, 2008
MathSciNet review: 2399032
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Abstract: Let $ K$ be an algebraically closed field of characteristic 0, and let $ E$ be the infinite dimensional Grassmann (or exterior) algebra over $ K$. Denote by $ P_n$ the vector space of the multilinear polynomials of degree $ n$ in $ x_1$, ..., $ x_n$ in the free associative algebra $ K(X)$. The symmetric group $ S_n$ acts on the left-hand side on $ P_n$, thus turning it into an $ S_n$-module. This fact, although simple, plays an important role in the theory of PI algebras since one may study the identities satisfied by a given algebra by applying methods from the representation theory of the symmetric group. The $ S_n$-modules $ P_n$ and $ KS_n$ are canonically isomorphic. Letting $ A_n$ be the alternating group in $ S_n$, one may study $ KA_n$ and its isomorphic copy in $ P_n$ with the corresponding action of $ A_n$. Henke and Regev described the $ A_n$-codimensions of the Grassmann algebra $ E$, and conjectured a finite generating set of the $ A_n$-identities for $ E$. Here we answer their conjecture in the affirmative.


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Additional Information

Dimas José Gonçalves
Affiliation: IMECC, UNICAMP, Cx. P. 6065, 13083-970 Campinas, SP, Brazil
Email: dimasjg@ime.unicamp.br

Plamen Koshlukov
Affiliation: IMECC, UNICAMP, Cx. P. 6065, 13083-970 Campinas, SP, Brazil
Email: plamen@ime.unicamp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09281-2
PII: S 0002-9939(08)09281-2
Received by editor(s): April 23, 2007
Received by editor(s) in revised form: June 19, 2007
Published electronically: April 3, 2008
Additional Notes: The first author was supported by a Ph.D. grant from PRPG, UNICAMP
The second author was partially supported by grants from CNPq (No. 302655/2005-0), and from FAPESP (No. 2004/13766-2 and 2005/60337-2)
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.