56,876 research outputs found
Extended partial order and applications to tensor products
We extend the preorder on k-tuples of dominant weights of a simple complex
Lie algebra g of classical type adding up to a fixed weight defined
by V. Chari, D. Sagaki and the author. We show that the induced extended
partial order on the equivalence classes has a unique minimal and a unique
maximal element. For k=2 we compute its size and determine the cover relation.
To each k-tuple we associate a tensor product of simple g-modules and we show
that for k=2 the dimension increases also along with the extended partial
order, generalizing a theorem proved in the aforementioned paper. We also show
that the tensor product associated to the maximal element has the biggest
dimension among all tuples for arbitrary k, indicating that this might be a
symplectic (resp. orthogonal) analogon of the row shuffle defined by Fomin et
al.
The extension of the partial order reduces the number of elements in the
cover relation and may facilitate the proof of an analogon of Schur positivity
along the partial order for symplectic and orthogonal types.Comment: 16 pages, final version, to appear in AJo
Schur Positivity and Kirillov-Reshetikhin Modules
In this note, inspired by the proof of the Kirillov-Reshetikhin conjecture,
we consider tensor products of Kirillov-Reshetikhin modules of a fixed node and
various level. We fix a positive integer and attach to each of its partitions
such a tensor product. We show that there exists an embedding of the tensor
products, with respect to the classical structure, along with the reverse
dominance relation on the set of partitions
Overview of Constrained PARAFAC Models
In this paper, we present an overview of constrained PARAFAC models where the
constraints model linear dependencies among columns of the factor matrices of
the tensor decomposition, or alternatively, the pattern of interactions between
different modes of the tensor which are captured by the equivalent core tensor.
Some tensor prerequisites with a particular emphasis on mode combination using
Kronecker products of canonical vectors that makes easier matricization
operations, are first introduced. This Kronecker product based approach is also
formulated in terms of the index notation, which provides an original and
concise formalism for both matricizing tensors and writing tensor models. Then,
after a brief reminder of PARAFAC and Tucker models, two families of
constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models,
are described in a unified framework, for order tensors. New tensor
models, called nested Tucker models and block PARALIND/CONFAC models, are also
introduced. A link between PARATUCK models and constrained PARAFAC models is
then established. Finally, new uniqueness properties of PARATUCK models are
deduced from sufficient conditions for essential uniqueness of their associated
constrained PARAFAC models
A finite element approach for vector- and tensor-valued surface PDEs
We derive a Cartesian componentwise description of the covariant derivative
of tangential tensor fields of any degree on general manifolds. This allows to
reformulate any vector- and tensor-valued surface PDE in a form suitable to be
solved by established tools for scalar-valued surface PDEs. We consider
piecewise linear Lagrange surface finite elements on triangulated surfaces and
validate the approach by a vector- and a tensor-valued surface Helmholtz
problem on an ellipsoid. We experimentally show optimal (linear) order of
convergence for these problems. The full functionality is demonstrated by
solving a surface Landau-de Gennes problem on the Stanford bunny. All tools
required to apply this approach to other vector- and tensor-valued surface PDEs
are provided
Tensor Distributions in the Presence of Degenerate Metrics
Tensor distributions and their derivatives are described without assuming the
presence of a metric. This provides a natural framework for discussing tensor
distributions on manifolds with degenerate metrics, including in particular
metrics which change signature.Comment: REVTeX, 19 pages; submitted to IJMP
Higher Derivative Gravity and Torsion from the Geometry of C-spaces
We start from a new theory (discussed earlier) in which the arena for physics
is not spacetime, but its straightforward extension-the so called Clifford
space (-space), a manifold of points, lines, areas, etc..; physical
quantities are Clifford algebra valued objects, called polyvectors. This
provides a natural framework for description of supersymmetry, since spinors
are just left or right minimal ideals of Clifford algebra. The geometry of
curved -space is investigated. It is shown that the curvature in -space
contains higher orders of the curvature in the underlying ordinary space. A
-space is parametrized not only by 1-vector coordinates but also by
the 2-vector coordinates , 3-vector coordinates , etc., called also {\it holographic coordinates}, since they
describe the holographic projections of 1-lines, 2-loops, 3-loops, etc., onto
the coordinate planes. A remarkable relation between the "area" derivative \p/
\p \sigma^{\mu \nu} and the curvature and torsion is found: if a scalar valued
quantity depends on the coordinates this indicates the
presence of torsion, and if a vector valued quantity depends so, this implies
non vanishing curvature. We argue that such a deeper understanding of the
-space geometry is a prerequisite for a further development of this new
theory which in our opinion will lead us towards a natural and elegant
formulation of -theory.Comment: 19 pages; A section describing the main physical implications of
C-space is added, and the rest of the text is modified accordingl
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