Symmetric closure

About: Symmetric closure is a research topic. Over the lifetime, 647 publications have been published within this topic receiving 16038 citations.

Differential Geometry and Symmetric Spaces

Abstract: Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces On the classification of symmetric spaces Functions on symmetric spaces Bibliography List of notational conventions Symbols frequently used Author index Subject index Reviews for the first edition.

The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions

Abstract: Group Representations.- Representations of the Symmetric Group.- Combinatorial Algorithms.- Symmetric Functions.- Applications and Generalizations.

Geometric Analysis on Symmetric Spaces

Abstract: A duality in integral geometry A duality for symmetric spaces The fourier transform on a symmetric space The Radon transform on $X$ and on $X_o$. Range questions Differential equations on symmetric spaces Eigenspace representations Solutions to exercises Bibliography Symbols frequently used Index.

Symmetric Tensors and Symmetric Tensor Rank

Abstract: A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order-$k$ tensor is the outer product of $k$ nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of which is symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz [J. Algebraic Geom., 4 (1995), pp. 201-222], is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most $r$ is not closed unless $r=1$.