Related papers: Orbifold E-functions of dual invertible polynomial…

A. Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito…

The semisimple Frobenius manifolds related to the Hurwitz spaces $H_{g,N}(k_1, ..., k_l)$ are considered. We show that the corresponding isomonodromic tau-function $\tau_I$ coincides with $(-1/2)$-power of the Bergmann tau-function which…

Ehrhart polynomials are extensively-studied structures that interpolate the discrete volume of the dilations of integral $n$-polytopes. The coefficients of Ehrhart polynomials, however, are still not fully understood, and it is not known…

We associate to a semisimple complex Lie algebra $\mathfrak{g}$ a sequence of polynomials $P_{\ell,\mathfrak{g}}(x)\in\mathbb{Q}[x]$ in $r$ variables, where $r$ is the rank of $\mathfrak{g}$ and $\ell=0,1,2,\ldots $. The polynomials…

We define the monoidal category $(Poly_E,y,\triangleleft)$ of polynomials under composition in any category $E$ with finite limits, including both cartesian and vertical morphisms of polynomials, and generalize to this setting the Dirichlet…

Motivated by the work of Koornwinder, Macdonald, Cherednik, Noumi, and van Diejen we define a 6-parameter double affine Hecke algebra and establish its basic structural properties, including the existence of an involution. We relate the…

We show that if a compact connected $n$-dimensional manifold $M$ has a conformal class containing two non-homothetic metrics $g$ and $\tilde g=e^{2\varphi}g$ with non-generic holonomy, then after passing to a finite covering, either $n=4$…

Orthogonal polynomials for a family of weight functions on $[-1,1]^2$, $$ \CW_{\a,\b,\g}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} (1-x^2)^\g(1-y^2)^{\g}, $$ are studied and shown to be related to the Koornwinder polynomials defined on the region…

Given a vector bundle $\mathcal E$ on a connected compact complex manifold $X$, [FLS] use a notion of completed Hochschild homology $\hat{\text{HH}}$ of $\text{Diff}(\mathcal E)$ such that $\hat{\text{HH}}_0(\text{Diff}(\mathcal E))$ is…

Let $F\in\mathbb{C}[x,y]$ be a polynomial, $\gamma(z)\in \pi_1(F^{-1}(z))$ a non-trivial cycle in a generic fiber of $F$ and let $\omega$ be a polynomial $1$-form, thus defining a polynomial deformation $dF+\epsilon\omega=0$ of the…

A $\mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided-difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this…

We study E-eigenvalues of a symmetric tensor $f$ of degree $d$ on a finite-dimensional Euclidean vector space $V$, and their relation with the E-characteristic polynomial of $f$. We show that the leading coefficient of the E-characteristic…

In this work we start by determining all irreducible spherical functions $\Phi$ of any $K $-type associated to the pair $(G,K)=(\SO(4),\SO(3))$. The functions $P=P(u)$ corresponding to the irreducible spherical functions of a fixed $K$-type…

Unstable operations in a generalized cohomology theory E give rise to a functor from the category of algebras over E to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this…

We introduce the theory of local and global monodromies of polynomials in cohomology groups in various geometric situations, focusing on its relations with toric geometry and motivic Milnor fibers, and moreover in the modern languages of…

The modified Macdonald polynomials introduced by Garsia and Haiman (1996) have many remarkable combinatorial properties. One such class of properties involves applying the $\nabla$ operator of Bergeron and Garsia (1999) to basic symmetric…

Given a flat metric one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local bi-Hamiltonian structure, and with additional…

We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between…

Calogero-Sutherland models associated to the Weyl groups of type A and B with exchange terms included in the Hamiltonians systems have non-symmetric eigenfunctions, which are products of the ground state with members of a family of…

The ring of dual numbers over a ring $R$ is $R[\alpha] = R[x]/(x^2)$, where $\alpha$ denotes $x+(x^2)$. For any finite commutative ring $R$, we characterize null polynomials and permutation polynomials on $R[\alpha]$ in terms of the…