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This paper considers matrix convex sets invariant under several types of rotations. It is known that matrix convex sets that are free semialgebraic are solution sets of Linear Matrix Inequalities (LMIs); they are called free spectrahedra. We classify all free spectrahedra that are circular, that is, closed under multiplication by e i t : up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal. A matrix convex set is called free circular if it is closed under left multiplication by unitary matrices. As a consequence of a Hahn–Banach separation theorem for free circular matrix convex sets, we show the coefficients of a minimal LMI defining a free circular free spectrahedron have, up to unitary equivalence, a block decomposition as above with only two blocks. This paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix. Up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials. [ABSTRACT FROM AUTHOR]

In the study of the variety of nilpotent elements in a Lie algebra, Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. In this paper, with the assumption that the ground field is algebraically closed of characteristic p > 3, we confirm this conjecture for the Lie algebras of Cartan type S˜ n and Sn. Moreover, we show that the variety of nilpotent elements in Sn is a complete intersection. Motivated by the proof of the irreducibility, we describe explicitly the ring of invariant polynomial functions on Sn. [ABSTRACT FROM AUTHOR]

Abstract: Fourth root metrics are a special and important class of Finsler metrics, which have been applied to physics. In this paper, we study invariant fourth root Finsler metrics on the Grassmannian manifolds . By using the results from the theory of invariant polynomials of Lie groups, we obtain a complete classification of such metrics. Further, some invariant -th root Finsler metrics are also given. [ABSTRACT FROM AUTHOR]

Abstract: A polynomial invariant under the action of a finite group can be rewritten using generators of the invariant ring. We investigate the complexity aspects of this rewriting process; we show that evaluation techniques enable one to reach a polynomial cost. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to derive quadrature estimates on compact, homogeneous manifolds embedded in Euclidean spaces, via energy functionals associated with a class of group-invariant kernels which are generalizations of zonal kernels on the spheres or radial kernels in euclidean spaces. Our results apply, in particular, to weighted Riesz kernels defined on spheres and certain projective spaces. Our energy functionals describe both uniform and perturbed uniform distribution of quadrature point sets. [Copyright &y& Elsevier]

There are some phenomena arising in the virtual knot theory which are not the case for classical knots. One of them deals with the "breaking" procedure of knots and obtaining long knots. Unlike the classical case, they might not be the same. The present work is devoted to construction of some invariants of long virtual links. Several explicit examples are given. For instance, we show how to prove the non-triviality of some knots obtained by breaking virtual unknot diagrams by very simple means. [ABSTRACT FROM AUTHOR]