Multiplication Theorem Research Articles

A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties

We prove a type-uniform Chevalley formula for multiplication with divisor classes in the equivariant quantum $K$-theory ring of any cominuscule flag variety $G/P$. We also prove that multiplication with divisor classes determines the equivariant quantum $K$-theory of arbitrary flag varieties. These results prove a conjecture of Gorbounov and Korff concerning the equivariant quantum $K$-theory of Grassmannians of Lie type A.

Universality of the Periodic Hurwitz Zeta-Function with Rational Parameter

The periodic Hurwitz zeta-function, a generalization of the classical Hurwitz zeta-function, is defined by a Dirichlet series with periodic coefficients and depends on a fixed parameter. We show that a wide class of analytic functions is approximated by shifts of a periodic zeta-function with rational parameter.

Joint discrete universality for periodic zeta-functions

In the paper, joint universality theorems for periodic zeta functions with multiplicative coefficients and periodic Hurwitz zeta-functions are proved. The main theorem of [11] is extended, and two new joint universality theorems on the approximation of a collection of analytic functions by discrete shifts of the above zeta-functions are obtained. For this, certain linear independence hypotheses are applied.

A new family of Lerch-type zeta functions interpolating a certain class of higher-order Apostol-type numbers and Apostol-type polynomials

The aim of this paper is to investigate some classes of higher-order Apostol-type numbers and Apostol-type polynomials. We construct Lerch-type zeta functions which interpolate these numbers and polynomials at negative integers. Moreover, by combining some well-known identities such as the Chu-Vandermonde identity with the Lerch-type zeta functions and generating functions for the higher- order Apostol-type numbers and Apostol-type polynomials, we derive some relations and identities including functional equation for these Lerch-type zeta functions with other zeta type functions, Raabe-type multiplication formula for the higher-order Apostol-type polynomials and the Stirling numbers. Finally, we give some remarks and observations on Lerch-type zeta functions and their functional equations.

A Multiplication Formula for the Modified Caldero\u2013Chapoton Map

A frieze in the modern sense is a map from the set of objects of a triangulated category mathsf {C} to some ring. A frieze X is characterised by the property that if tau xrightarrow yrightarrow x is an Auslander–Reiten triangle in mathsf {C}, then X(tau x)X(x)-X(y)=1. The canonical example of a frieze is the (original) Caldero–Chapoton map, which send objects of cluster categories to elements of cluster algebras. Holm and Jørgensen (Nagoya Math J 218:101–124, 2015; Bull Sci Math 140:112–131, 2016), the notion of generalised friezes is introduced. A generalised frieze X' has the more general property that X'(tau x)X'(x)-X'(y)in {0,1}. The canonical example of a generalised frieze is the modified Caldero–Chapoton map, also introduced in Holm and Jørgensen (2015, 2016). Here, we develop and add to the results in Holm and Jørgensen (2016). We define Condition F for two maps alpha and beta in the modified Caldero–Chapoton map, and in the case when mathsf {C} is 2-Calabi–Yau, we show that it is sufficient to replace a more technical “frieze-like” condition from Holm and Jørgensen (2016). We also prove a multiplication formula for the modified Caldero–Chapoton map, which significantly simplifies its computation in practice.

Multiplication Formulas and Canonical Bases for Quantum Affine gln

AbstractWe will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra of a cyclic quiver Δ(n). As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established by J. Y. Guo and C. M. Ringel, and derive a recursive formula to compute them. We will further use the formula and the construction of a monomial basis for given by Deng, Du, and Xiao together with the double Ringel-Hall algebra realisation of the quantum loop algebra given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for . As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most 2 for the quantum group .

MULTIPLICATION FORMULAS AND SEMISIMPLICITY FOR -SCHUR SUPERALGEBRAS

We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for the $q$-Schur superalgebras. This gives a combinatorialization of the relative norm approach developed in Du and Gu (A realization of the quantum supergroup$\mathbf{U}(\mathfrak{g}\mathfrak{l}_{m|n})$, J. Algebra 404 (2014), 60–99). We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for $q$-Schur superalgebras. We also construct infinitesimal and little $q$-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.

On weak peak quasisymmetric functions

In this paper, we construct the weak version of peak quasisymmetric functions inside the Hopf algebra of weak composition quasisymmetric functions (WCQSym) defined by Guo, Thibon and Yu. Weak peak quasisymmetric functions (WPQSym) are studied in several aspects. First we find a natural basis of WPQSym lifting peak functions introduced by Stembridge. Then we confirm that WPQSym is a Hopf subalgebra of WCQSym by giving explicit multiplication, comultiplication and antipode formulas. By extending Stembridge's descent-to-peak maps, we also show that WPQSym is a Hopf quotient of WCQSym. On the other hand, we prove that WPQSym embeds as a Rota–Baxter subalgebra of WCQSym, thus of the free commutative Rota–Baxter algebra of weight 1 on one generator. Moreover, WPQSym can also be a Rota–Baxter quotient of WCQSym.

Torsion des variétés abéliennes CM

In this note, we improve on a result of Silverberg giving an upper bound for the order of a rational torsion point on a CM abelian variety over a number field in terms of the degree of the field and the dimension of the variety. The proof uses the main theorem of complex multiplication and class field theory.

An identity of symmetry for the degenerate Frobenius-Euler Polynomials

Abstract In this paper, we prove an identity of symmetry for the higher-order degenerate Frobenius-Euler polynomials and derive the recurrence relations and multiplication theorem type result for the degenerate Frobenius-Euler polynomials.

Plancherel theorem and quaternion Fourier transform for square integrable functions

ABSTRACTThe quaternion Fourier transform (QFT), a generalization of the classical 2D Fourier transform, plays an increasingly active role in particular signal and colour image processing. There tends to be an inordinate degree of interest placed on the properties of QFT. The classical convolution theorem and multiplication formula are only suitable for 2D Fourier transform of complex-valued signal, and do not hold for QFT of quaternion-valued signal. The purpose of this paper is to overcome these problems and establish the Plancherel and inversion theorems of QFT in the square integrable signals space . First, we investigate the behaviours of QFT in the integrable signals space . Next, we deduce the energy preservation property which extends functions from to space. Moreover, some other important properties such as modified multiplication formula are also analyzed for QFT.

On some saturated numerical semigroups with multiplicity eight

In this paper, we will investigate saturated numerical semigroups with multiplicity 8 and conductor C. Also, we will give formulas for Frobenius number, determiner number and genus of these semigroups.

Expansion formulas for Apostol type Q-Appell polynomials, and their special cases

We present identities of various kinds for generalized $q$-Apostol-Bernoulli and Apostol-Euler polynomials and power sums, which resemble $q$-analogues of formulas from the 2009 paper by Liu and Wang. These formulas are divided into two types: formulas with only $q$-Apostol-Bernoulli, and only $q$-Apostol-Euler polynomials, or so-called mixed formulas, which contain polynomials of both kinds.This can be seen as a logical consequence of the fact that the $q$-Appell polynomials form a commutative ring. The functional equations for Ward numbers operating on the $q$-exponential function, as well as symmetry arguments, are essential for many of the proofs.We conclude by finding multiplication formulas for two $q$-Appell polynomials of general form. This brings us to the $q$-H polynomials, which were discussed in a previous paper.

The BLM realization for the integral form of Uv(glm|n)

In this paper, we embed the integral form of the quantum supergroup ${\boldsymbol~U}_\bv(\mathfrak{gl}_{m\,|\,n})$ to the product of a family of integral quantum Schur superalgebras. We show that the image of the embedding is a free $\mathbb{Z}[{\boldsymbol~v},{\boldsymbol~v}^{-1}]$-module by finding the basis explicitly and calculating the fundamental multiplication formulas of these bases. Unlike the non-super case, the fundamental multiplication formula, which is the key step, is more complicated since we have to deal with the case of multiplying the odd root vectors. As a consequence, via the base change, we realize the quantum supergroup at roots of unity as a subalgebra of the product of quantum Schur superalgebras. Thus, we find a new basis of quantum supergroups at odd roots of unity which comes from quantum Schur superalgebras.

BLM realization for 𝒰ℤ(𝔤𝔩 ̂n)

In 1990, Beilinson–Lusztig–MacPherson (BLM) discovered a realization for quantum [Formula: see text] via a geometric setting of quantum Schur algebras. We will generalize their result to the classical affine case. More precisely, we first use Ringel–Hall algebras to construct an integral form [Formula: see text] of [Formula: see text], where [Formula: see text] is the universal enveloping algebra of the loop algebra [Formula: see text]. We then establish the stabilization property of multiplication for the classical affine Schur algebras. This stabilization property leads to the BLM realization of [Formula: see text] and [Formula: see text]. In particular, we conclude that [Formula: see text] is a [Formula: see text]-Hopf subalgebra of [Formula: see text]. As a bonus, this method leads to an explicit [Formula: see text]-basis for [Formula: see text], and it yields explicit multiplication formulas between generators and basis elements for [Formula: see text]. As an application, we will prove that the natural algebra homomorphism from [Formula: see text] to the affine Schur algebra over [Formula: see text] is surjective.

A WEIGHTED DISCRETE UNIVERSALITY THEOREM FOR PERIODIC ZETA-FUNCTIONS. II

In the paper, a weighted theorem on the approximation of a wide class of analytic functions by shifts ζ(s + ikαh; a), k ∈ N, 0 < α < 1, and h > 0, of the periodic zeta-function ζ(s; a) with multiplicative periodic sequence a, is obtained.

Gromov–Witten invariants of the Hilbert schemes of points of a K3 surface

We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface. Let $S$ be a K3 surface and let $\mathsf{Hilb}^d(S)$ be the Hilbert scheme of $d$ points of $S$. In case of elliptically fibered K3 surfaces $S \to \mathbb{P}^1$, we calculate genus $0$ Gromov-Witten invariants of $\mathsf{Hilb}^d(S)$, which count rational curves incident to two generic fibers of the induced Lagrangian fibration $\mathsf{Hilb}^d(S) \to \mathbb{P}^d$. The generating series of these invariants is the Fourier expansion of a power of the Jacobi theta function times a modular form, hence of a Jacobi form. We also prove results for genus $0$ Gromov-Witten invariants of $\mathsf{Hilb}^d(S)$ for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov-Witten invariants of the Hilbert scheme of $2$ points of $\mathbb{P}^1 \times E$, where $E$ is an elliptic curve. Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on $\mathsf{Hilb}^d(S)$ with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of $S$. We prove the conjecture in the first non-trivial case $\mathsf{Hilb}^2(S)$. As a corollary, we find that the full genus $0$ Gromov-Witten theory of $\mathsf{Hilb}^2(S)$ in primitive classes is governed by Jacobi forms. We present two applications. A conjecture relating genus $1$ invariants of $\mathsf{Hilb}^d(S)$ to the Igusa cusp form was proposed in joint work with R. Pandharipande in \cite{K3xE}. Our results prove the conjecture in case $d=2$. Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through $2$ general points.

Formules de genres et conjecture de Greenberg

Greenberg’s well known conjecture, (GC) for short, asserts that the Iwasawa invariants $$\lambda $$ and $$\mu $$ associated to the cyclotomic $${\mathbb {Z}}_p$$ -extension of any totally real number field F should vanish. In his foundational 1976 paper, Greenberg has shown two necessary and sufficient conditions for (GC) to hold, in two seemingly opposite cases, when p is undecomposed, resp. totally decomposed in F. In this article we present an encompassing approach covering both cases and resting only on “ genus formulas ”, that is (roughly speaking) on formulas which express the order of the Galois (co-)invariants of certain modules along the cyclotomic tower. These modules are akin to class groups, and in the end we obtain several unified criteria, which naturally contain the particular conditions given by Greenberg.

Four dimentional joint moments due to Dirichlet density and their applications in summability of quadruple hypergeometric functions

Using the Exton’s multiple joint moments in four dimensional spaces due to Dirichlet density and a generalization of Bosanquet and Kestelman theorem , we prove some theorems in summability of the series containing quadruple hypergeomtric functions. These theorems generalize some well known generating functions and multiplication theorems involving product of hypergeometric functions of one and more variables. We discuss some other applications and establish several interesting particular cases. Finally, we obtain an approximation formula of the series involving Exton’s quadruple hypergeometric function.

Gröbner-Shirshov basis for degenerate Ringel-Hall algebra of type $$D_4$$

In this paper, we first give a presentation of the degenerate Ringel-Hall algebra $$\mathfrak {H}_0(D_4)$$ by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra. Then, by using the relations which are computed to give this presentation, we construct the Grobner-Shirshov basis for this algebra.