-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 | References | Similar Articles | Additional Information

Abstract: Let be an algebraically closed field of characteristic 0, and let be the infinite dimensional Grassmann (or exterior) algebra over . Denote by the vector space of the multilinear polynomials of degree in , ..., in the free associative algebra . The symmetric group acts on the left-hand side on , thus turning it into an -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 -modules and are canonically isomorphic. Letting be the alternating group in , one may study and its isomorphic copy in with the corresponding action of . Henke and Regev described the -codimensions of the Grassmann algebra , and conjectured a finite generating set of the -identities for . Here we answer their conjecture in the affirmative.

**1.**A. Henke and A. Regev,*Explicit decompositions of the group algebras 𝐹𝑆_{𝑛} and 𝐹𝐴_{𝑛}*, Polynomial identities and combinatorial methods (Pantelleria, 2001), Lecture Notes in Pure and Appl. Math., vol. 235, Dekker, New York, 2003, pp. 329–357. MR**2021806****2.**A. Henke and A. Regev,*𝐴-codimensions and 𝐴-cocharacters*, Israel J. Math.**133**(2003), 339–355. MR**1968434**, 10.1007/BF02773073**3.**D. Krakowski and A. Regev,*The polynomial identities of the Grassmann algebra*, Trans. Amer. Math. Soc.**181**(1973), 429–438. MR**0325658**, 10.1090/S0002-9947-1973-0325658-5

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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

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.