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Defining relations of low degree of invariants of two 4 x 4 matrices
No full textThe trace algebra C_nd over a field of characteristic 0 is generated by all traces of products of d generic n × n matrices, n, d ≥ 2. Minimal sets of generators of C_nd are known for n = 2 and 3 for any d and for n = 4 and 5 and d = 2. The explicit defining relations between the generators are found for n = 2 and any d and for n = 3, d = 2 only. Defining relations of minimal degree for n = 3 and any d are also known. The minimal degree of the defining relations of any homogeneous minimal generating set of C_42 is equal to 12. Starting with the generating set given recently by Drensky and Sadikova, we have determined all relations of degree ≤ 14. For this purpose we have developed further algorithms based on representation theory of the general linear group and easy computer calculations with standard functions of Maple
Grobner bases of ideals invariant under endomorphisms
No full textWe introduce the notion of Gröbner S-basis of an ideal of the free associative algebra K over a field K invariant under the action of a semigroup S of endomorphisms of the algebra. We calculate the Gröbner S-bases of the ideal corresponding to the universal enveloping algebra of the free nilpotent of class 2 Lie algebra and of the T-ideal generated by the polynomial identity [x, y, z] = 0, with respect to suitable semigroups S. In the latter case, if |X | > 2, the ordinary Gröbner basis is infinite and our Gröbner S-basis is finite. We obtain also explicit minimal Gröbner bases of these ideals
Defining Relations of Noncommutative Trace Algebra of Two 3 x 3 Matrices
No full textThe noncommutative (or mixed) trace algebra Tnd is generated by d generic n × n matrices and by the
algebra Cnd generated by all traces of products of generic matrices, n, d 2. It is known that over a field
of characteristic 0 this algebra is a finitely generated free module over a polynomial subalgebra S of the
center Cnd. For n = 3 and d = 2 we have found explicitly such a subalgebra S and a set of free generators
of the S-module T32.We give also a set of defining relations of T32 as an algebra and a Gröbner basis of the
corresponding ideal. The proofs are based on easy computer calculations with standard functions of Maple,
the explicit presentation of C32 in terms of generators and relations, and methods of representation theory
of the general linear grou
Defining relations of minimal degree of the trace algebra of 3 X 3 matrices
Get PDFThe trace algebra Cnd over a field of characteristic 0 is generated by all traces
of products of d generic n × n matrices, n, d 2. Starting with the generating set of
C3d given by Abeasis and Pittaluga in 1989, we have shown that the minimal degree
of the set of defining relations of C3d is equal to 7 for any d 3. We have determined
all relations of minimal degree. For d = 3 we have also found the defining relations
of degree 8
Computing with rational symmetric functions and applications to invariant theory and PI-algebras
Get PDF2010 Mathematics Subject Classification: 05A15, 05E05, 05E10, 13A50, 15A72, 16R10, 16R30, 20G05Let K be a field of any characteristic. Let the formal power series f(x1, ..., xd) = ∑ αnx1^n1 ··· xd^nd = ∑ m(λ)Sλ(x1, ..., xd), αn, m(λ) ∈ K, be a symmetric function decomposed as a series of Schur functions. When f is a rational function whose denominator is a product of binomials of the form 1−x1^a1 ··· xd^ad, we use a classical combinatorial method of Elliott of 1903
further developed in the Ω-calculus (or Partition Analysis) of MacMahon in 1916 to compute the generating function X
M(f;x1, ..., xd ) = ∑ m(λ)x1^λ1 ··· xd^λd, λ = (λ1, ..., λd). M is a rational function with denominator of a similar form as f. We apply the method to several problems on symmetric algebras, as well as problems in classical invariant theory, algebras with polynomial identities, and noncommutative invariant theory.The research of the first named author was partially supported by INdAM. The research of the second, third, and fourth named authors was partially supported by Grant for Bilateral Scientific Cooperation between Bulgaria and Ukraine. The research of the fifth named author was partially supported by NSF Grant DMS-1016086
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