Related papers: On spherical designs obtained from $Q$-polynomial …

Let $Y$ denote a $D$-class symmetric association scheme with $D \geq 3$, and suppose $Y$ is almost-bipartite P- and Q-polynomial. Let $x$ denote a vertex of $Y$ and let $T=T(x)$ denote the corresponding Terwilliger algebra. We prove that…

It is known that there is a close analogy between "Euclidean t-designs vs. spherical t-designs" and "Relative t-designs in binary Hamming association schemes vs. combinatorial t-designs". In this paper, we want to prove how much we can…

We develop skein theory for 3-manifolds in the presence of codimension-one defects, focusing especially on defects arising from parabolic induction/restriction for quantum groups. We use these defects as a model for the quantum decorated…

In this work we study the automorphisms of skew $PBW$ extensions and skew quantum polynomials. We use Artamonov's works as reference for getting the principal results about automorphisms for generic skew $PBW$ extensions and generic skew…

Let $K_n$ be a complete graph with $n$ vertices. An embedding of $K_n$ in $S^3$ is called a spatial $K_n$-graph. Knots in a spatial $K_n$-graph corresponding to simple cycles of $K_n$ are said to be constituent knots. We consider the case…

We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of…

We provide a combinatorial description of the coefficients appearing in the expansion of Hall-Littlewood polynomials in terms of monomial symmetric functions. We also give a Littlewood-Richardson rule for Hall-Littlewood polynomials. For…

The decomposition of the polynomials on the quaternionic unit sphere in $\Hd$ into irreducible modules under the action of the quaternionic unitary (symplectic) group and quaternionic scalar multiplication has been studied by several…

Intersection numbers for subspace designs are introduced and $q$-analogs of the Mendelsohn and K\"ohler equations are given. As an application, we are able to determine the intersection structure of a putative $q$-analog of the Fano plane…

We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by…

Unitary t-designs are some of the most versatile tools in quantum information theory. Their applications range from randomized benchmarking and shadow tomography, to more fundamental ones such as emulating quantum chaos and establishing…

The characteristic polynomial plays an important role in study of hyperplane arrangements. There are several refinements of the characteristic polynomial. One of them is the coboundary polynomial defined by Crapo. Another refinement is the…

In this summary paper, we present the key ideas behind the recent proof of the $K(\pi, 1)$ conjecture for affine Artin groups, which states that complements of locally finite affine hyperplane arrangements with real equations and stable…

Graphical designs are an extension of spherical designs to functions on graphs. We connect linear codes to graphical designs on cube graphs, and show that the Hamming code in particular is a highly effective graphical design. We show that…

A spherical $t$-design is a set of points on the sphere that are nodes of a positive equal weight quadrature rule having algebraic accuracy $t$ for all spherical polynomials with degrees $\le t$. Spherical $t$-designs have many…

For a field $E$ of characteristic different from $2$ and cohomological $2$-dimension one, quadratic forms over the rational function field $E(X)$ are studied. A characterisation in terms of polynomials in $E[X]$ is obtained for having that…

2-Dim quantum Poincare` Group E_q(1,1) at roots of unity, its dual U_q(e(1,1)) and some of its homogeneous spaces are introduced. Invariant integrals on E_q(1,1) and its invariant discrete subgroup E(1,1\mid p) are constructed.…

Let $K$ be a field and $\sigma$ an automorphism of $K$ of order $n$.Employing a nonassociative algebra, we study the eigenspace of a bounded skew polynomial $f\in K[t;\sigma]$. We mainly treat the case that $K/F$ is a cyclic field extension…

The geometry of spaces with indefinite inner product, known also as Krein spaces, is a basic tool for developing Operator Theory therein. In the present paper we establish a link between this geometry and the algebraic theory of…

In terms of the creative microscoping method recently introduced by Guo and Zudilin [Adv. Math. 346 (2019), 329--358], we find a $q$-supercongruence with four parameters modulo $\Phi_n(q)(1-aq^n)(a-q^n)$, where $\Phi_n(q)$ denotes the…