$A$-identities for the Grassmann algebra: The conjecture of Henke and Regev
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- by Dimas José Gonçalves and Plamen Koshlukov PDF
- Proc. Amer. Math. Soc. 136 (2008), 2711-2717 Request permission
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.References
- 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
- A. Henke and A. Regev, $A$-codimensions and $A$-cocharacters, Israel J. Math. 133 (2003), 339–355. MR 1968434, DOI 10.1007/BF02773073
- 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
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
- MR Author ID: 105210
- Email: plamen@ime.unicamp.br
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2711-2717
- MSC (2000): Primary 16R10; Secondary 16R40, 16R50
- DOI: https://doi.org/10.1090/S0002-9939-08-09281-2
- MathSciNet review: 2399032