44,919 research outputs found
Tensor rank is not multiplicative under the tensor product
The tensor rank of a tensor t is the smallest number r such that t can be
decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an
l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is
sub-multiplicative under the tensor product. We revisit the connection between
restrictions and degenerations. A result of our study is that tensor rank is
not in general multiplicative under the tensor product. This answers a question
of Draisma and Saptharishi. Specifically, if a tensor t has border rank
strictly smaller than its rank, then the tensor rank of t is not multiplicative
under taking a sufficiently hight tensor product power. The "tensor Kronecker
product" from algebraic complexity theory is related to our tensor product but
different, namely it multiplies two k-tensors to get a k-tensor.
Nonmultiplicativity of the tensor Kronecker product has been known since the
work of Strassen.
It remains an open question whether border rank and asymptotic rank are
multiplicative under the tensor product. Interestingly, lower bounds on border
rank obtained from generalised flattenings (including Young flattenings)
multiply under the tensor product
Entropy Games and Matrix Multiplication Games
Two intimately related new classes of games are introduced and studied:
entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on
a finite arena by two-and-a-half players: Despot, Tribune and the
non-deterministic People. Despot wants to make the set of possible People's
behaviors as small as possible, while Tribune wants to make it as large as
possible.An MMG is played by two players that alternately write matrices from
some predefined finite sets. One wants to maximize the growth rate of the
product, and the other to minimize it. We show that in general MMGs are
undecidable in quite a strong sense.On the positive side, EGs correspond to a
subclass of MMGs, and we prove that such MMGs and EGs are determined, and that
the optimal strategies are simple. The complexity of solving such games is in
NP\&coNP.Comment: Accepted to STACS 201
Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group
We inspect the relationship between relative Fourier multipliers on
noncommutative Lebesgue-Orlicz spaces of a discrete group and relative
Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four
applications are given: lacunary sets; unconditional Schauder bases for the
subspace of a Lebesgue space determined by a given spectrum, that is, by a
subset of the group; the norm of the Hilbert transform and the Riesz projection
on Schatten-von-Neumann classes with exponent a power of 2; the norm of
Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less
than 1.Comment: Corresponds to the version published in the Canadian Journal of
Mathematics 63(5):1161-1187 (2011
Cluster algebras via cluster categories with infinite-dimensional morphism spaces
We apply our previous work on cluster characters for Hom-infinite cluster
categories to the theory of cluster algebras. We give a new proof of
Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky's Cluster
algebras IV for skew-symmetric cluster algebras. We also construct an explicit
bijection sending certain objects of the cluster category to the decorated
representations of Derksen, Weyman and Zelevinsky, and show that it is
compatible with mutations in both settings. Using this map, we give a
categorical interpretation of the E-invariant and show that an arbitrary
decorated representation with vanishing E-invariant is characterized by its
g-vector. Finally, we obtain a substitution formula for cluster characters of
not necessarily rigid objects.Comment: 32 pages, added referenc
- …