Some Kazhdan–Lusztig Coefficients of Affine Weyl Group of Type B~ 2

Let [Formula: see text] be the affine Weyl group of type [Formula: see text], on which we consider the length function [Formula: see text] from [Formula: see text] to [Formula: see text] and the Bruhat order [Formula: see text]. For [Formula: see text] in [Formula: see text], let [Formula: see text] be the coefficient of [Formula: see text] in Kazhdan–Lusztig polynomial [Formula: see text]. We determine some [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] is the lowest two-sided cell of [Formula: see text] and [Formula: see text] is the higher one. Furthermore, we get some consequences using left or right strings and some properties of leading coefficients.

Symmetric polynomials in the variety generated by Grassmann algebras

Let [Formula: see text] denote the variety generated by infinite-dimensional Grassmann algebras, i.e. the collection of all unitary associative algebras satisfying the identity [Formula: see text], where [Formula: see text]. Consider the free algebra [Formula: see text] in [Formula: see text] generated by [Formula: see text]. We call a polynomial [Formula: see text] symmetric if it is preserved under the action of the symmetric group [Formula: see text] on generators, i.e. [Formula: see text] for each permutation [Formula: see text]. The set of symmetric polynomials forms the subalgebra [Formula: see text] of invariants of the group [Formula: see text] in [Formula: see text]. The commutator ideal [Formula: see text] of the algebra [Formula: see text] has a natural left [Formula: see text]-module structure, and [Formula: see text] is a left [Formula: see text]-module. We give a finite free generating set for the [Formula: see text]-module [Formula: see text].

On μ-symmetric polynomials

We study functions of the roots of an integer polynomial [Formula: see text] with [Formula: see text] distinct roots [Formula: see text] of multiplicity [Formula: see text], [Formula: see text]. Traditionally, root functions are studied via the theory of symmetric polynomials; we generalize this theory to [Formula: see text]-symmetric polynomials. We initiate the study of the vector space of [Formula: see text]-symmetric polynomials of a given degree [Formula: see text] via the concepts of [Formula: see text]-gist and [Formula: see text]-ideal. In particular, we are interested in the root function [Formula: see text]. The D-plus discriminant of [Formula: see text] is [Formula: see text]. This quantity appears in the complexity analysis of the root clustering algorithm of Becker et al. (ISSAC 2016). We conjecture that [Formula: see text] is [Formula: see text]-symmetric, which implies [Formula: see text] is rational. To explore this conjecture experimentally, we introduce algorithms for checking if a given root function is [Formula: see text]-symmetric. We design three such algorithms: one based on Gröbner bases, another based on canonical bases and reduction, and the third based on solving linear equations. Each of these algorithms has variants that depend on the choice of a basis for the [Formula: see text]-symmetric functions. We implement these algorithms (and their variants) in Maple and experiments show that the latter two algorithms are significantly faster than the first.

On monogenity of certain pure number fields defined by xpr − m

Let [Formula: see text] be a pure number field generated by a complex root [Formula: see text] of a monic irreducible polynomial [Formula: see text] where [Formula: see text] is a square free rational integer, [Formula: see text] is a rational prime integer, and [Formula: see text] is a positive integer. In this paper, we study the monogenity of [Formula: see text]. We prove that if [Formula: see text], then [Formula: see text] is monogenic. But if [Formula: see text] and [Formula: see text], then [Formula: see text] is not monogenic. Some illustrating examples are given.

Symmetric polynomials of algebras related with 2 × 2 generic traceless matrices

Let [Formula: see text] and [Formula: see text] be two generic traceless matrices of size [Formula: see text] with entries from a commutative associative polynomial algebra over a field [Formula: see text] of characteristic zero. Consider the associative unitary algebra [Formula: see text], and its Lie subalgebra [Formula: see text] generated by [Formula: see text] and [Formula: see text] over the field [Formula: see text]. It is well known that the center [Formula: see text] of [Formula: see text] is the polynomial algebra generated by the algebraically independent commuting elements [Formula: see text], [Formula: see text], [Formula: see text]. We call a polynomial [Formula: see text] symmetric, if [Formula: see text]. The set of symmetric polynomials is equal to the algebra [Formula: see text] of invariants of symmetric group [Formula: see text]. Similarly, we define the Lie algebra [Formula: see text] of symmetric polynomials in the Lie algebra [Formula: see text]. We give the description of the algebras [Formula: see text] and [Formula: see text], and we provide finite sets of free generators for [Formula: see text], and [Formula: see text] as [Formula: see text]-modules.

The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n − 1

In this paper, we study the irreducible quotient [Formula: see text] of the polynomial representation of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] over an algebraically closed field of positive characteristic [Formula: see text] where [Formula: see text]. In the [Formula: see text] case, for all [Formula: see text] we give a complete description of the polynomials in the maximal proper graded submodule [Formula: see text], the kernel of the contravariant form [Formula: see text], and subsequently find the Hilbert series of the irreducible quotient [Formula: see text]. In the [Formula: see text] case, we give a complete description of the polynomials in [Formula: see text] when the characteristic [Formula: see text] and [Formula: see text] is transcendental over [Formula: see text], and compute the Hilbert series of the irreducible quotient [Formula: see text]. In doing so, we prove a conjecture due to Etingof and Rains completely for [Formula: see text], and also for any [Formula: see text] and [Formula: see text]. Furthermore, for [Formula: see text], we prove a simple criterion to determine whether a given polynomial [Formula: see text] lies in [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text] fixed.

On nonlocal perturbation of the problem on eigenvalues of differentiation operator on a segment

This work is devoted to the construction of a characteristic polynomial of the spectral problem of a first-order differential equation on an interval with a spectral parameter in a boundary value condition with integral perturbation which is an entire analytic function of the spectral parameter. Based on the characteristic polynomial formula, conclusions about the asymptotics of the spectrum of the perturbed spectral problem are established.

More on the unimodality of domination polynomial of a graph

Let [Formula: see text] be a graph of order [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a dominating set of [Formula: see text] if every vertex in [Formula: see text] is adjacent to at least one vertex of [Formula: see text]. The domination polynomial of [Formula: see text] is the polynomial [Formula: see text], where [Formula: see text] is the number of dominating sets of [Formula: see text] of size [Formula: see text], and [Formula: see text] is the size of a smallest dominating set of [Formula: see text], called the domination number of [Formula: see text]. Motivated by a conjecture in [S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, ARS Combin. 114 (2014) 257–266] which states that the domination polynomial of any graph is unimodal, we obtain sufficient conditions for this conjecture to hold. Also we study the unimodality of graph [Formula: see text] with [Formula: see text], where [Formula: see text] is an integer.

Optimum Angle of Force Production Temporarily Changes Due to Growth in Male Adolescence

The peak increase in lean mass in adolescents is delayed from peak height velocity (PHV), and muscle flexibility temporarily decreases as bones grow. If the decrease in muscle flexibility is caused by muscle elongation, the relationship between the exerted torque and the joint angle could change in adolescents. The purpose of this study was to investigate the change in the optimum angle of force production due to growth. Eighty-eight healthy boys were recruited for this study. Isokinetic knee extension muscle strength of the dominant leg was recorded. The outcome variable was the knee flexion angle when maximal knee extension torque was produced (optimum angle). The age at which PHV occurred was estimated from subjects’ height history. We calculated the difference between the age at measurement and the expected age of PHV (growth age). A regression analysis was performed with the optimal angle of force exertion as the dependent variable and the growth age as the independent variable. Then, a polynomial formula with the lowest p-value was obtained. A significant cubic regression was obtained between optimum angle and growth age. The results suggest that the optimum angle of force production temporarily changes in male adolescence.

Characterizing some polarized Fano fibrations via Hilbert curves

The Hilbert curve of a complex polarized manifold [Formula: see text] is the complex affine plane curve of degree [Formula: see text] defined by the Hilbert-like polynomial [Formula: see text], where [Formula: see text] is the canonical bundle of [Formula: see text] and [Formula: see text] and [Formula: see text] are regarded as complex variables. A natural expectation is that this curve encodes several properties of the pair [Formula: see text]. In particular, the existence of a fibration of [Formula: see text] over a variety of smaller dimension induced by a suitable adjoint bundle to [Formula: see text] translates into the fact that the Hilbert curve has a quite special shape. Along this line, Hilbert curves of special varieties like Fano manifolds with low coindex, as well as fibrations over low-dimensional varieties having such a manifold as general fiber, endowed with appropriate polarizations, are investigated. In particular, several polarized manifolds relevant for adjunction theory are completely characterized in terms of their Hilbert curves.