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In this paper, we find a lower bound for the order of the group 〈 θ + α 〉 ⊂ F ‾ q ⁎ , where α ∈ F q , θ is a generic root of the polynomial F A , r ( X ) = b X q r + 1 − a X q r + d X − c ∈ F q [ X ] and a d − b c ≠ 0 . [ABSTRACT FROM AUTHOR]

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]

The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules S λ. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases—essentially, only the cases λ = (d , d) and λ = (d , d , d). Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the 2-parameter family of Specht modules with λ of the form (d , d , 1 ℓ). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics. [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]

. This article considers the problem of generating polynomial invariants for iterative loops with loop initialization statements and nonsingular linear operators in loop bodies. The set of such invariants forms an ideal in the ring of polynomials in the loop variables. Two algorithms are presented one of which calculates basic invariants for a linear operator in the form of a Jordan cell and the other calculates basic invariants for a diagonalizable linear operator with an irreducible minimal characteristic polynomial. The following theorem on the structure of the basis of the ideal of invariants for such an operator is proved: this basis consists of basic invariants of Jordan cells and basic invariants of the diagonalizable part of the linear operator being considered. [ABSTRACT FROM AUTHOR]

Let G be a real reductive Lie group and H a closed subgroup of G which is reductive in G. In our earlier work it was shown that, if the homomorphism i : H·(gC; hC;C) → H·(gC; (kH)C;C) is not injective, there does not exist a compact manifold locally modelled on G/H. In this paper, we give a classification of the semisimple symmetric spaces G/H for which i is not injective. We also study the case when G cannot be realised as a linear group. [ABSTRACT FROM AUTHOR]

In this paper, we propose a matrix-variate generalization of the Gauss hypergeometric distribution and study several of its properties. We also derive probability density functions of the product of two independent random matrices when one of them is Gauss hypergeometric. These densities are expressed in terms of Appell’s first hypergeometric function F1 and Humbert’s confluent hypergeometric function Φ1of matrix arguments. [ABSTRACT FROM AUTHOR]

Abstract: Let R be a prime ring, C its extended centroid and (resp. Q) its left (resp. symmetric) Martindale quotient ring. Let δ be a σ-derivation of R, where σ is an automorphism of R. We show the equivalence of K-polynomials (resp. K-identities) of δ and cv-polynomials (resp. semi-invariant polynomials) in the Ore extension . We prove the existence of K-polynomials of δ in certain rather general family of maps. As applications, the following are proved among other things: Consider the expression , where and . [(1)] If then either R is a GPI-ring or for all . [(2)] If then either R is commutative or for all . [Copyright &y& Elsevier]

Abstract: We study the factorization of polynomials of the form over the finite field . We show that these polynomials are closely related to a natural action of the projective linear group on non-linear irreducible polynomials over . Namely, irreducible factors of are exactly those polynomials that are invariant under the action of some non-trivial element . This connection enables us to enumerate irreducibles which are invariant under . Since the class of polynomials includes some interesting polynomials like or , our work generalizes well-known asymptotic results about the number of irreducible polynomials and the number of self-reciprocal irreducible polynomials over . At the same time, we generalize recent results about certain invariant polynomials over the binary field . [Copyright &y& Elsevier]

Abstract: In this article, we derive several properties such as marginal distribution, moments involving zonal polynomials, and asymptotic expansion of the complex bimatrix variate beta type 1 distribution introduced by Dı´az-Garcı´a and Gutiérrez Jáimez [José A. Dı´az-Garcı´a, Ramón Gutiérrez Jáimez, Complex bimatrix variate generalised beta distributions, Linear Algebra Appl. 432 (2010) 571–582]. We also derive distributions of several matrix valued functions of random matrices jointly distributed as complex bimatrix variate beta type 1. [Copyright &y& Elsevier]

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: Alternating matrix polynomials, that is, polynomials whose coefficients alternate between symmetric and skew-symmetric matrices, generalize the notions of even and odd scalar polynomials. We investigate the Smith forms of alternating matrix polynomials, showing that each invariant factor is an even or odd scalar polynomial. Necessary and sufficient conditions are derived for a given Smith form to be that of an alternating matrix polynomial. These conditions allow a characterization of the possible Jordan structures of alternating matrix polynomials, and also lead to necessary and sufficient conditions for the existence of structure-preserving strong linearizations. Most of the results are applicable to singular as well as regular matrix polynomials. [Copyright &y& Elsevier]

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]