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The focus of this work is a family of maps from the space of symmetric matrices, , into the space for any , invariant under the conjugate action of the orthogonal group . This family, called the generated k-isotropic functions, generalizes known types of maps with a similar invariance property, such as the spectral, isotropic, primary matrix functions, multiplicative compound, and additive compound matrices on . We give necessary and sufficient conditions for the analyticity of these maps. [ABSTRACT FROM AUTHOR]

This article gives a new matrix function named ‘twisted immanant,’ which can be regarded as an analogue of the immanant. This is defined for each self-conjugate partition through a ‘twisted’ analogue of the irreducible character of the symmetric group. This twisted immanant has some interesting properties. For example, it satisfies Cauchy–Binet type formulas. Moreover it is closely related to the following results for matrices whose entries anticommute with each other: (i) the description of the invariants under the conjugations, and (ii) an analogue of the Cauchy identities for symmetric polynomials. [ABSTRACT FROM AUTHOR]

Closed expressions for matrix elements 〈lm'|A(G)|lm〉, where |lm〉 are spherical functions and A(G) is the average of all symmetry operators of point group G, are derived for all point groups (PGs) and then used to obtain linear combinations of spherical functions that are totally symmetric under all symmetry operations of G. In the derivation, we exploit the product structure of the groups. The obtained expressions are used to explore properties of multipoles of symmetric charge distributions. We produce complete lists of selection rules for multipoles Ql and their moments Qlm, as well as of numbers of independent moments in a multipole, for any l and m and for all PGs. Periodicities and other trends in these properties are revealed. [ABSTRACT FROM AUTHOR]

Abstract: We propose a method for diagonalizing matrices with entries in commutative rings. The point of departure is to split the characteristic polynomial of the matrix over a (universal) splitting algebra, and to use the resulting universal roots to construct eigenvectors of the matrix. A crucial point is to determine when the determinant of the eigenvector matrix, that is the matrix whose columns are the eigenvectors, is regular in the splitting algebra. We show that this holds when the matrix is generic, that is, the entries are algebraically independent over the base ring. It would have been desirable to have an explicit formula for the determinant in the generic case. However, we have to settle for such a formula in a special case that is general enough for proving regularity in the general case. We illustrate the uses of our results by proving the Spectral Mapping Theorem, and by generalizing a fundamental result from classical invariant theory. [Copyright &y& Elsevier]

Abstract: We consider lower-triangular matrices consisting of symmetric polynomials, and we show how to factorize and invert them. Since binomial coefficients and Stirling numbers can be represented in terms of symmetric polynomials, these results contain factorizations and inverses of Pascal and Stirling matrices as special cases. This work generalizes that of several other authors on Pascal and Stirling matrices. [Copyright &y& Elsevier]

By means of matrix decomposition and the partial fraction method, we establish several determinant evaluation formulas, which can be considered as generalizations of the Vandermonde and Cauchy determinants. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the author gives a simpler alternative approach to the LU factorization and 1-banded factorization of the Vandermonde matrix, and obtains explicit formulas of the triangular factors and 1-banded matrices by using symmetric functions. [Copyright &y& Elsevier]