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We propose and investigate a family of maps from the space of n × n symmetric matrices, Sn, into the space Snk for any k = 1, . . . ,n which are invariant under the conjugate action of the orthogonal group On. This family, called k-isotropic functions, not only generalizes all known types of maps with similar invariance property, such as the spectral, isotropic, primary matrix functions, multiplicative compound, and additive compound matrices on Sn, but also contains many previously unknown and potentially interesting subclasses of functions. We give necessary and sufficient conditions for these maps to be r-times differentiable and exhibit a recursive formula for the rth derivative. Moreover, we provide a full description of the linear k-isotropic functions which provides a generalization to constitutive equation in classical linear elasticity also know as Hooke law. Finally, we exhibit a duality motivated by the Hodge star operator between different classes of k-isotropic functions. The study of isotropic maps is motivated by constitutive relations in continuum mechanics, where the properties of materials and their symmetry groups are related. [ABSTRACT FROM AUTHOR]

We apply down operators in the afine nilCoxeter algebra to yield explicit combinatorial expansions for certain families of noncommutative κ-Schur functions. This yields a combinatorial interpretation for a new family of κ-Littlewood-Richardson coefficients. [ABSTRACT FROM AUTHOR]

Real valued functions, F(X), on a symmetric matrix argument are called spectral if F(UTXU) = F(X) for every orthogonal matrix U and X ϵ dom F. We are interested in a description of the higher order derivatives (when they exist) of F with respect to X. Formulae for the gradient and the Hessian of F are given in [A. S. Lewis, Math. Oper. Res., 21 (1996), pp. 576–588] and [A. S. Lewis and H. S. Sendov, SIAM Matrix Anal. Appl., 23 (2001), pp. 368–386]. In this work we present common features of these two formulae that are preserved in the higher order derivatives. [ABSTRACT FROM AUTHOR]

Any spectral function can be written as a composition of a symmetric function f : Rn → R and the eigenvalue function λ(·) : S → Rn, often denoted by (f ○ λ), where S is the subspace of n × n symmetric matrices. In this paper, we present some nonsmooth analysis for such spectral functions. Our main results are (a) (f ○ λ) is directionally differentiable if f is semidifferentiable, (b) (f ○ λ) is LC1 if and only if f is LC1, and (c) (f ○ λ) is SC1 if and only if f is SC1. Result (a) is complementary to a known (negative) fact that (f ○ λ) might not be directionally differentiable if f is directionally differentiable only. Results (b) and (c) are particularly useful for the solution of LC1 and SC1 minimization problems which often can be solved by fast (generalized) Newton methods. Our analysis makes use of recent results on continuously differentiable spectral functions as well as on nonsmooth symmetric­matrix-valued functions. [ABSTRACT FROM AUTHOR]