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Papers by Francesca Benanti

$Research paper thumbnail of Defining Relations of Minimal Degree of the Trace Algebra of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">3 \times 3</annotation></semantics></math>3×3 Matrices$

The trace algebra C(n,d) over a field of characteristic 0 is generated by all traces of products ... more The trace algebra C(n,d) over a field of characteristic 0 is generated by all traces of products of d generic nxn matrices, n,d>1. Minimal sets of generators of C(n,d) are known for n=2 and n=3 for any d as well as for n=4 and n=5 and d=2. The defining relations between the generators are found for n=2 and any d

We consider associative PI-algebras over a field of characteristic zero. We study the asymptotic ... more We consider associative PI-algebras over a field of characteristic zero. We study the asymptotic behavior of the sequence of multiplicities of the cocharacters for some significant classes of algebras. We also give a char- acterization of finitely generated algebras for which this behavior is linear or quadratic.

We present a survey on polynomial identities of matrices over a field of characteristic 0 from co... more 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 iden- tities of matrices and related objects. Among the other applications, these methods have been successfully used:

Linear Algebra and its Applications, 2000

Let M3(F) be the algebra of 3×3 matrices with involution * over a field F of characteristic zero.... more Let M3(F) be the algebra of 3×3 matrices with involution * over a field F of characteristic zero. We study the ∗-polynomial identities of M3(F), where ∗=t is the transpose involution, through the representation theory of the hyperoctahedral group Bn. After decomposing the space of multilinear ∗-polynomial identities of degree n under the Bn-action, we determine which irreducible Bn-modules appear

Journal of Algebra, 2008

The trace algebra C nd over a field of characteristic 0 is generated by all traces of products of... more The 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 n = 3 for any d as well as for n = 4 and n = 5 and d = 2. The defining relations between the generators are found for n = 2 and any d and for n = 3, d = 2 only. Starting with the generating set of C 3d given by Abeasis and Pittaluga in 1989, we have shown that the minimal degree of the set of defining relations of C 3d 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. The proofs are based on methods of representation theory of the general linear group and easy computer calculations with standard functions of Maple.

Journal of Algebra, 2003

Let A be a finitely generated superalgebra over a field F of characteristic 0. To the graded poly... more Let A be a finitely generated superalgebra over a field F of characteristic 0. To the graded polynomial identities of A one associates a numerical sequence {cnsup(A)}n⩾1 called the sequence of graded codimensions of A. In case A satisfies an ordinary polynomial identity, such sequence is exponentially bounded and we capture its exponential growth by proving that for any such algebra limn→∞cnsup(A)n

Israel Journal of Mathematics, 2006

We consider associative P I-algebras over a field of characteristic zero. The main goal of the pa... more We consider associative P I-algebras over a field of characteristic zero. The main goal of the paper is to prove that the codimensions of a verbally prime algebra are asymptotically equal to the codimensions of the T -ideal generated by some Amitsur's Capelli-type polynomials E *

Communications in Algebra, 1998

ABSTRACT

$Research paper thumbnail of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>∗-Subvarieties of the Variety Generated by <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true" stretchy="true" minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>M</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">K</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>t</mi><mo fence="true" stretchy="true" minsize="1.2em" maxsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\bigl( M_2(\mathbb{K}),t \bigr)</annotation></semantics></math>(M2(K),t)$

Canadian Journal of Mathematics, 2003

Advances in Applied Mathematics, 2006

The noncommutative (or mixed) trace algebra T nd is generated by d generic n × n matrices and by ... more The noncommutative (or mixed) trace algebra T nd is generated by d generic n × n matrices and by the algebra C nd 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 C nd . For n = 3 and d = 2 we have found explicitly such a subalgebra S and a set of free generators of the S-module T 32 . We give also a set of defining relations of T 32 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 C 32 in terms of generators and relations, and methods of representation theory of the general linear group.

Arxiv preprint arXiv: …, 2012