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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]

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: 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]