Generic 2 × 2 matrices in positive characteristic

Asparouhov, Tihomir; Drensky, Vesselin; Koev, Plamen; Tsiganchev, Dimitar

doi:10.1006/jabr.1999.8143

Cited by 4 publications

(7 citation statements)

References 15 publications

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“…Given this discrepancy between characteristic 0 and characteristic p > 0, we would like to study affine algebras, at least in the relatively free case, by passing modulo p. Unfortunately this cannot be done naively, due to counterexamples of Schelter [21] and Asparouhov-DrenskyKoev-Tsiganchev [2]. But the idea does work for large enough p. Thus, taking p as in the theorem, one could solve the isomorphism problem for two relatively free PI-algebras.…”

Section: Corollary 21mentioning

confidence: 94%

Normal bases of PI-algebras

Kanel-Belov

Rowen

Vishne

2006

Advances in Applied Mathematics

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Normal bases of affine PI-algebras are studied through the following stages: essential height, monomial algebras, representability, and modular reduction.In this survey we review the algorithmic theory of PI-algebras, in terms of normal bases, and indicate directions for further research. In view of Kemer [15], one can study normal bases in terms of the codimension theory of PI-algebras, of which Regev is the pioneer. This topic is developed in [10,19]. Thus we feel this paper is appropriate for a volume honoring Regev.Let A be an associative affine algebra over an infinite field k, generated by the set Ω = {a 1 , . . . , a }. Ordering the letters a 1 < · · · < a induces the lexicographic order on the set Ω * of words in the generators over the alphabet: w < v if |w| < |v|, or if |w| = |v| and w is lexicographically smaller than v. The normal base of the algebra A with respect to the ordered set Ω, is the set of all words in Ω * that cannot be written as a linear combination of smaller words [4,9,24]. Obviously this is a base of A (as a vector space).

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Section: Corollary 21mentioning

confidence: 94%

Normal bases of PI-algebras

Kanel-Belov

Rowen

Vishne

2006

Advances in Applied Mathematics

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“…A version of the above scheme was used by Schelter [71] to find the first new polynomial identities for 2 × 2 matrices over a field of characteristic 2 which do not originate from the identities of M 2 (Z) and the identity 2x = 0. See also [28] for a multilinear version of this result (obtained by calculations by hand) and [2].…”

Section: Algebraic and Combinatorial Backgroundmentioning

confidence: 96%

“…Once we have at our disposal an essential weak polynomial identity, the method of Razmyslov gives a central polynomial. In this way the existence of the weak identity [x 2 1 , x 2 ] of degree three for M 2 (K) gives rise to a central polynomial of degree four. Similarly the central polynomials for M n (K) of degree n 2 constructed by Halpin [39] used weak polynomial identities of degree n(n + 1)/2.…”

Section: The Main Objectsmentioning

confidence: 99%

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Computational Approach to Polynomial Identities of Matrices — a Survey

Benanti¹

2003

Polynomial Identities and Combinatorial Methods

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We present a survey on polynomial identities of matrices over a field of characteristic 0 from computational point of view. We describe several computational methods for calculation with polynomial identities of matrices and related objects. Among the other applications, these methods have been successfully used: S 9-characters of *-polynomial identities of degree 9 in symmetric variables only for 6 × 6 matrices with symplectic involution.

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“…i on top of it; the obtained tableau is standard. 1 2 n w x Using the terminology of 10, 11 , one says that the polynomial ² :…”

Section: ž P Q I Jmentioning

confidence: 99%