The polynomial identities of the Grassmann algebra

Krakowski, Don; Regev, Amitai

doi:10.1090/s0002-9947-1973-0325658-5

Cited by 77 publications

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“…It was also proved in [3] that T (E) is generated as a T -ideal by the single polynomial [x 1 , x 2 , x 3 ], and a detailed description of the module structure of the S n -module P n (E) = P n /(T (E) ∩ P n ) was obtained as well. Furthermore one of the main results in [2] states that c …”

Section: Introductionmentioning

confidence: 99%

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

Gonçalves¹,

Koshlukov

2008

Proc. Amer. Math. Soc.

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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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Section: Introductionmentioning

confidence: 99%

“…It was proved in [3] that for the Grassmann algebra one has c n (E) = 2 n−1 . It was also proved in [3] that T (E) is generated as a T -ideal by the single polynomial [x 1 , x 2 , x 3 ], and a detailed description of the module structure of the S n -module P n (E) = P n /(T (E) ∩ P n ) was obtained as well.…”

Section: Introductionmentioning

confidence: 99%

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

Gonçalves¹,

Koshlukov

2008

Proc. Amer. Math. Soc.

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“…Coming back to the Grassmann algebra E, we know that in the ordinary case T (E) is generated by the triple commutator [x, y, z] := [[x, y], z] as shown in the papers by Latyshev [27] and [26] by Krakovski and Regev. For this purpose, we want to point out that the latter two papers deal with characteristic 0 only even if the argument used in [27] is still valid in positive characteristic as the argument used in [17] Theorem 5.1.2.…”

Section: Introductionmentioning

confidence: 99%

On the graded identities of the Grassmann algebra

Centrone¹

2017

Journal of Algebra Combinatorics Discrete Structures and Applications

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“…The polynomial identities for G were described in [9] by Krakowski and Regev when char(F )=0, and by various authors in the general case (see [4] and [10]). The central polynomials for the Grassmann algebra were described independently by several authors, see for example [1], [2] and [6].…”

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confidence: 99%

“…Its dimension, c n (R), is called the n-th codimension of R. The codimensions of G were computed explicitly in [9].…”

mentioning

confidence: 99%