scholarly journals ON TENSOR PRODUCT DECOMPOSITION OF $k$-TRIDIAGONAL TOEPLITZ MATRICES

2015 ◽  
Vol 103 (3) ◽  
Author(s):  
A. Ohashi ◽  
T.S. Usuda ◽  
T. Sogabe ◽  
F. Yilmaz
Keyword(s):  
Tensor Product ◽  
Toeplitz Matrices ◽  
Product Decomposition

scholarly journals An inequality of Kostka numbers and Galois groups of Schubert problems

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Christopher J. Brooks ◽  
Abraham Mart\'ın Campo ◽  
Frank Sottile
Keyword(s):  
Spectral Analysis ◽  
Fourier Series ◽  
Tensor Product ◽  
Toeplitz Matrices ◽  
Galois Groups ◽  
Alternating Group ◽  
Special Position ◽  
Produit Tensoriel ◽  
Kostka Numbers ◽  
International Audience

International audience We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. Using a criterion of Vakil and a special position argument due to Schubert, this follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, an easy combinatorial injection proves the inequality. For the remaining cases, we use that these Kostka numbers appear in tensor product decompositions of $\mathfrak{sl}_2\mathbb{C}$ -modules. Interpreting the tensor product as the action of certain commuting Toeplitz matrices and using a spectral analysis and Fourier series rewrites the inequality as the positivity of an integral. We establish the inequality by estimating this integral. On montre que le groupe de Galois de tout problème de Schubert concernant des droites dans l'espace projective contient le groupe alterné. En utilisant un critère de Vakil et l'argument de position spéciale due à Schubert, ce résultat se déduit d'une inégalité particulière des nombres de Kostka des tableaux ayant deux rangées. Dans la plupart des cas, une injection combinatoriale facile montre l’inégalité. Pour les cas restants, on utilise le fait que ces nombres de Kostka apparaissent dans la décomposition en produit tensoriel des $\mathfrak{sl}_2\mathbb{C}$-modules. En interprétant le produit tensoriel comme l'action de certaines matrices de Toeplitz commutant entre elles, et en utilisant de l'analyse spectrale et les séries de Fourier, on réécrit l’inégalité comme la positivité d'une intégrale. L’inégalité sera établie en estimant cette intégrale.

scholarly journals Invariant and Coinvariant Spaces for the Algebra of Symmetric Polynomials in Non-Commuting Variables

10.37236/438 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
François Bergeron ◽  
Aaron Lauve
Keyword(s):  
Tensor Product ◽  
Symmetric Group ◽  
Symmetric Polynomials ◽  
Reflection Groups ◽  
Invariant Polynomials ◽  
Product Decomposition

We analyze the structure of the algebra $\mathbb{K}\langle\mathbf{x}\rangle^{\mathfrak{S}_n}$ of symmetric polynomials in non-commuting variables in so far as it relates to $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$, its commutative counterpart. Using the "place-action" of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of $\mathbb{K}\langle\mathbf{x}\rangle^{\mathfrak{S}_n}$ analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups. Résumé. Nous analysons la structure de l'algèbre $\mathbb{K}\langle\mathbf{x}\rangle^{\mathfrak{S}_n}$ des polynômes symétriques en des variables non-commutatives pour obtenir des analogues des résultats classiques concernant la structure de l'anneau $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$ des polynômes symétriques en des variables commutatives. Plus précisément, au moyen de "l'action par positions", on réalise $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$ comme sous-module de $\mathbb{K}\langle\mathbf{x}\rangle^{\mathfrak{S}_n}$. On découvre alors une nouvelle décomposition de $\mathbb{K}\langle\mathbf{x}\rangle^{\mathfrak{S}_n}$ comme produit tensorial, obtenant ainsi un analogues des théorèmes classiques de Chevalley et Shephard-Todd.

scholarly journals A full variational calculation based on a tensor product decomposition

1989 ◽  
Vol 160 (4) ◽  
pp. 423-431
Author(s):  
Frederick A. Senese ◽  
Christopher A. Beattie ◽  
John C. Schug ◽  
Jimmy W. Viers ◽  
Layne T. Watson
Keyword(s):  
Tensor Product ◽  
Variational Calculation ◽  
Product Decomposition

scholarly journals VON NEUMANN ENTROPY AND RELATIVE POSITION BETWEEN SUBALGEBRAS

2013 ◽  
Vol 24 (08) ◽  
pp. 1350066 ◽  
Author(s):  
MARIE CHODA
Keyword(s):  
Tensor Product ◽  
Relative Position ◽  
Von Neumann Algebra ◽  
Positive Definite ◽  
The Other ◽  
Von Neumann Entropy ◽  
Product Decomposition ◽  
Positive Definite Matrices ◽  
Von Neumann ◽  
Finite Von Neumann Algebra

In order to give numerical characterizations of the notion of "mutual orthogonality", we introduce two kinds of family of positive definite matrices for a unitary u in a finite von Neumann algebra M. They are arising from u naturally depending on the decompositions of M. One corresponds to the tensor product decomposition and the other does to the crossed product decomposition. By using the von Neumann entropy for these positive definite matrices, we characterize the notion of mutual orthogonality between subalgebras.

Tensor Product Decomposition, Entanglement, and Bogoliubov Transformations for Two Fermion System

2005 ◽  
Vol 12 (02) ◽  
pp. 179-188 ◽  
Author(s):  
Paweł Caban ◽  
Krzysztof Podlaski ◽  
Jakub Rembieliński ◽  
Kordian A. Smoliński ◽  
Zbigniew Walczak
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Explicit Formula ◽  
Fermion System ◽  
Density Matrices ◽  
Product Decomposition ◽  
Creation And Annihilation Operators ◽  
Separable States ◽  
Entanglement Of Formation ◽  
Bogoliubov Transformations

We consider the two-fermion system whose states are subjected to the superselection rule forbidding the superposition of states with fermionic and bosonic statistics. This implies that separable states are described only by diagonal density matrices. Moreover, we find the explicit formula for the entanglement of formation, which in this case cannot be calculated properly using Wootters's concurrence. We also discuss the problem of the choice of tensor product decomposition in a system of two fermions with the help of Bogoliubov transformations of creation and annihilation operators. Finally, we show that there exist states which are separable with respect to all tensor product decompositions of the underlying Hilbert space.

scholarly journals Entanglement and tensor product decomposition for two fermions

2005 ◽  
Vol 38 (6) ◽  
pp. L79-L86 ◽  
Author(s):  
P Caban ◽  
K Podlaski ◽  
J Rembielinski ◽  
K A Smolinski ◽  
Z Walczak
Keyword(s):  
Tensor Product ◽  
Product Decomposition
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