898 research outputs found
On Degeneration of Tensors and Algebras
An important building block in all current asymptotically fast algorithms for matrix multiplication are tensors with low border rank, that is, tensors whose border rank is equal or very close to their size. To find new asymptotically fast algorithms for matrix multiplication, it seems to be important to understand those tensors whose border rank is as small as possible, so called tensors of minimal border rank.
We investigate the connection between degenerations of associative algebras and degenerations of their structure tensors in the sense of Strassen. It allows us to describe an open subset of n*n*n tensors of minimal border rank in terms of smoothability of commutative algebras. We describe the smoothable algebra associated to the Coppersmith-Winograd tensor and prove a lower bound for the border rank of the tensor used in the "easy construction" of Coppersmith and Winograd
A note on the gap between rank and border rank
We study the tensor rank of the tensor corresponding to the algebra of
n-variate complex polynomials modulo the dth power of each variable. As a
result we find a sequence of tensors with a large gap between rank and border
rank, and thus a counterexample to a conjecture of Rhodes. At the same time we
obtain a new lower bound on the tensor rank of tensor powers of the generalised
W-state tensor. In addition, we exactly determine the tensor rank of the tensor
cube of the three-party W-state tensor, thus answering a question of Chen et
al.Comment: To appear in Linear Algebra and its Application
Generalized Matsumoto-Tits sections and quantum quasi-shuffle algebras
In this paper generalized Matsumoto-Tits sections lifting permutations to the
algebra associated to a generalized virtual braid monoid are defined. They are
then applied to study the defining relations of the quantum quasi-shuffle
algebras via the total symmetrization operator.Comment: 18 page
Tautological relations in Hodge field theory
We propose a Hodge field theory construction that captures algebraic
properties of the reduction of Zwiebach invariants to Gromov-Witten invariants.
It generalizes the Barannikov-Kontsevich construction to the case of higher
genera correlators with gravitational descendants.
We prove the main theorem stating that algebraically defined Hodge field
theory correlators satisfy all tautological relations. From this perspective
the statement that Barannikov-Kontsevich construction provides a solution of
the WDVV equation looks as the simplest particular case of our theorem. Also it
generalizes the particular cases of other low-genera tautological relations
proven in our earlier works; we replace the old technical proofs by a novel
conceptual proof.Comment: 35 page
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