arXiv subjects

Xiaoping Xu

Publications and source records attributed to Xiaoping Xu.

At least 19 recordsLinked to original sources

arXiv2026

Congruence Classes of Supporting the Erd\"{o}s-Straus Conjecture I: Tame Solutions

In 1948, Erd\"{o}s and Straus formulated a conjecture : for any positive integer $n>2$, there exist positive integers $n_1,n_2$ and $n_3$ such that \begin{equation}\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3},\nonumber\end{equation} which is still open. It is known that one only needs to prove the conjecture for any prime number $n$ such that $n\equiv 1\;(\mbox{mod}\;24)$. If $n=24m+1$ and $n_1\leq n_2,n_3$, then $n_1=6m+k$ with $1\leq k\leq 12m$. A solution $(n_1,n_2,n_3)$ of the above equation is called a {\it tame solution} if $n_2$ and $n_3$ are factors of $(6m+k)(24m+1)$. We call $n=24m+1$ {\it wild} if it does not have any tame solution. Computer calculation shows that there are only nine wild primes among the 7185 primes of the form $24m+1$ with $m\leq 30000$. In this paper, we derive the tame solutions of the above equation for the integers of the form $24m+1$ with $m$ parameterized by certain congruence classes. They cover the solvability of all the 586 tame primes among the 591 primes of the form $24m+1$ with $m\leq 2000$.

math.NT↗

arXiv2025

Hilbert Polynomials of Noncanonical Orthogonal Oscillator Representations of $sl(n)$

By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie algebras, Luo and the second author (2013) obtained a large family of infinite-dimensional irreducible representations of the algebras on the homogeneous solutions of the Laplace equation. In our earlier work, we proved that the associated varieties of these irreducible representations are the intersection of determinantal varieties. In this paper, we find the Hilbert polynomial $\mathfrak p_{ M}(k)$ of these associated varieties. Moreover, we show that the Hilbert polynomial $\mathfrak p_{ M, {M_0}}(k)$ of such an irreducible module $M$ with respect to any generating subspace $M_0$ satisfies $\mathfrak p_{_M}(k)\leq \mathfrak p_{ M, {M_0}}(k)$ for sufficiently large positive integer $k$ and find a necessary and sufficient condition that the equality holds. Furthermore, we explicitly determine the leading term of $\mathfrak p_{ M, {M_0}}(k)$, which is independent of the choice of $M_0$.

math.RT↗

arXiv2025

Orthogonal Oscillator Representations, Laplace Equations and Intersections of Determinantal Varieties

Associated varieties are geometric objects appearing in infinite-dimensional representations of semisimple Lie algebras (groups). By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie algebras, Luo and the second author (2013) obtained a big family of infinite-dimensional irreducible representations of the algebras on certain spaces of homogeneous solutions of the Laplace equation. In this paper, we prove that the associated varieties of these irreducible representations are the intersections of explicitly given determinantal varieties. This provides an explicit connection among representation theory, partial differential equations and algebraic geometry.

math.RT↗

arXiv2023

Full Conformal Oscillator Representations of Orthogonal Lie Algebras and Combinatorial Identities

Zhao and the second author (2013) constructed a functor from o(k)-Mod to o(k + 2)-Mod. In this paper, we use the functor successively to obtain an universal first-order differential operator realization for any highest-weight representation of o(2n + 3) in (n + 1)^2 variables and that of o(2n + 2) in n(n + 1) variables. When the highest weight is dominant integral, we determine the corresponding finite-dimensional irreducible module explicitly. One can use the result to study tensor decompositions of finite-dimensional irreducible modules by solving certain first-order linear partial differential equations, and thereby obtain the corresponding physically interested Clebsch-Gordan coefficients and exact solutions of Knizhnik-Zamolodchikov equation in WZW model of conformal field theory. We also find an equation of counting the dimension of an irreducible o(k + 2)-module in terms of certain alternating sum of the dimensions of irreducible o(k)-modules. In the case of the Steinberg modules, we obtain new combinatorial identities of classical type.

math.RT↗

arXiv2022

Full Projective Oscillator Representations of Special Linear Lie Algebras and Combinatorial Identities

Using the projective oscillator representation of sl(n+1) and Shen's mixed product for Witt algebras, Zhao and the second author (2011) constructed a new functor from sl(n)-Mod to sl(n+1)-Mod. In this paper, we start from n = 2 and use the functor successively to obtain a full projective oscillator realization of any finite-dimensional irreducible representation of sl(n+1). The representation formulas of all the root vectors of sl(n+1) are given in terms of first-order differential operators in n(n+1)/2 variables. One can use the result to study tensor decompositions of finite-dimensional irreducible modules by solving certain first-order linear partial differential equations, and thereby obtain the corresponding physically interested Clebsch-Gordan coefficients and exact solutions of Knizhnik-Zamolodchikov equation in WZW model of conformal field theory.

math.RT↗

arXiv2016

Representations of Lie Algebras and Partial Differential Equations

This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic codes, combinatorics and algebraic varieties. Various oscillator generalizations of the classical representation theorem on harmonic polynomials are presented. New functors from the representation category of a simple Lie algebra to that of another simple Lie algebra are given. Partial differential equations play key roles in solving certain representation problems. The weight matrices of the minimal and adjoint representations over the simple Lie algebras of types E and F are proved to generate ternary orthogonal linear codes with large minimal distances. New multi-variable hypergeometric functions related to the root systems of simple Lie algebras are introduced in connection with quantum many-body system in one-dimension. Certain equivalent combinatorial properties on representation formulas are found. Irreducibility of representations are proved directly related to algebraic varieties.

math.RT↗

arXiv2014

Conformal Oscillator Representations of Orthogonal Lie Algebras

The conformal transformations with respect to the metric defining the orthogonal Lie algebra o(n) give rise to a one-parameter (c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra o(n+2). Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector a, we prove that the space forms an irreducible o(n+2)-module for any constant c if the vector a is not on a certain hypersurface. By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of o(n+2) on the polynomial algebra C in n variables. Moreover, we prove that the algebra C forms an infinite-dimensional irreducible weight o(n+2)-module with finite-dimensional weight subspaces if the constant c is not a half integer.

math.RT↗

arXiv2014

Projective Oscillator Representations of sl(n+1) and sp(2m+2)

The n-dimensional projective group gives rise to a one-parameter family of inhomogeneous first-order differential operator representations of sl(n+1). By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of sl(n+1). Letting these differential operators act on the corresponding polynomial algebra and the space of exponential-polynomial functions, we construct new multi-parameter families of explicit infinite-dimensional irreducible representations for s(n+1) and sp(2m+2) when n=2m+1. Our results can be viewed as extensions of Howe's oscillator construction of infinite-dimensional multiplicity-free irreducible representations for sl(n).

math.RT↗

arXiv2013

Moving-Frame Approach to Nonlinear Internal Waves in Oceans

In this article, we introduce a moving-frame approach to the geophysical equation of two-dimensional uniformly stratified rotational fluid in oceans and find a family of exact solutions containing ten arbitrary parameter functions.

physics.flu-dyn↗

arXiv2012

Algebraic Approaches to Partial Differential Equations

Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by the author in recent years, with emphasis on physical equations such as: the Calogero-Sutherland model of quantum many-body system in one-dimension, the Maxwell equations, the free Dirac equations, the generalized acoustic system, the Kortweg and de Vries (KdV) equation, the Kadomtsev and Petviashvili (KP) equation, the equation of transonic gas flows, the short-wave equation, the Khokhlov and Zabolotskaya equation in nonlinear acoustics, the equation of geopotential forecast, the nonlinear Schrodinger equation and coupled nonlinear Schrodinger equations in optics, the Davey and Stewartson equations of three-dimensional packets of surface waves, the equation of the dynamic convection in a sea, the Boussinesq equations in geophysics, the incompressible Navier-Stokes equations and the classical boundary layer equations. In linear partial differential equations, we focus on finding all the polynomial solutions and solving the initial-value problems. Intuitive derivations of Lie symmetry of nonlinear partial differential equations are given. These symmetry transformations generate sophisticated solutions with more parameters from relatively simple ones. They are also used to simplify our process of finding exact solutions. We have extensively used moving frames, asymmetric conditions, stable ranges of nonlinear terms, special functions and linearizations in our approaches to nonlinear partial differential equations. The exact solutions we obtained usually contain multiple parameter functions and most of them are not of traveling-wave type.

math-ph↗

arXiv2012

Supersymmetyric Analogues of the Classical Theorem on Harmonic Polynomials

Classical harmonic analysis says that the spaces of homogeneous harmonic polynomials (solutions of Laplace equation) are irreducible modules of the corresponding orthogonal Lie group (algebra) and the whole polynomial algebra is a free module over the invariant polynomials generated by harmonic polynomials. In this paper, we first establish two-parameter $\mbb{Z}^2$-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebra $gl(n|m)$. Then we extend the result to two-parameter $\mbb{Z}$-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebras $osp(2n|2m)$ and $osp(2n+1|2m)$.

math.RT↗

arXiv2012

A New Factor from E6-Mod to E7-Mod

We find a new representation of the simple Lie algebra of type $E_7$ on the polynomial algebra in 27 variables, which gives a fractional representation of the corresponding Lie group on 27-dimensional space. Using this representation and Shen's idea of mixed product, we construct a functor from $E_6$-{\bf Mod} to $E_7$-{\bf Mod}. A condition for the functor to map a finite-dimensional irreducible $E_6$-module to an infinite-dimensional irreducible $E_7$-module is obtained. Our general frame also gives a direct polynomial extension from irreducible $E_6$-modules to irreducible $E_7$-modules. The obtained infinite-dimensional irreducible $E_7$-modules are $({\cal G},K)$-modules in terms of Lie group representations. The results could be used in studying the quantum field theory with $E_7$ symmetry and symmetry of partial differential equations.

math.RT↗

arXiv2011

A New Functor from $D_5$-Mod to $E_6$-Mod

We find a new representation of the simple Lie algebra of type $E_6$ on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen's idea of mixed product, we construct a functor from $D_5$-{\bf Mod} to $E_6$-{\bf Mod}. A condition for the functor to map a finite-dimensional irreducible $D_5$-module to an infinite-dimensional irreducible $E_6$-module is obtained. Our general frame also gives a direct polynomial extension from irreducible $D_5$-modules to irreducible $E_6$-modules. The obtained infinite-dimensional irreducible $E_6$-modules are $({\cal G},K)$-modules in terms of Lie group representations. The results could be used in studying the quantum field theory with $E_6$ symmetry and symmetry of partial differential equations.

math.RT↗

arXiv2011

Generalized Conformal Representations of Orthogonal Lie Algebras

The conformal transformations with respect to the metric defining $o(n,\mbb{C})$ give rise to a nonhomogeneous polynomial representation of $o(n+2,\mbb{C})$. Using Shen's technique of mixed product, we generalize the above representation to a non-homogenous representation of $o(n+2,\mbb{C})$ on the tensor space of any finite-dimensional irreducible $o(n,\mbb{C})$-module with the polynomial space, where a hidden central transformation is involved. Moreover, we find a condition on the constant value taken by the central transformation such that the generalized conformal representation is irreducible. In our approach, Pieri's formulas, invariant operators and the idea of Kostant's characteristic identities play key roles. The result could be useful in understanding higher-dimensional conformal field theory with the constant value taken by the central transformation as the central charge. Our representations virtually provide natural extensions of the conformal transformations on a Riemannian manifold to its vector bundles.

math.RT↗

arXiv2010

Oscillator Variations of the Classical Theorem on Harmonic Polynomials

We study two-parameter oscillator variations of the classical theorem on harmonic polynomials, associated with noncanonical oscillator representations of sl(n) and o(n). We find the condition when the homogeneous solution spaces of the variated Laplace equation are irreducible modules of the concerned algebras and the homogeneous subspaces are direct sums of the images of these solution subspaces under the powers of the dual differential operator. This establishes a local (sl(2),sl(n)) and (sl(2),o(n)) Howe duality, respectively. In generic case, the obtained irreducible o(n)-modules are infinite-dimensional non-unitary modules without highest-weight vectors. As an application, we determine the structure of noncanonical oscillator representations of sp(2n). When both parameters are equal to the maximal allowed value, we obtain an infinite family of explicit irreducible (G,K)-modules for o(n) and sp(2n). Methodologically we have extensively used partial differential equations to solve representation problems.

math.RT↗

arXiv2010

Generalized Projective Representations for sl(n+1)

It is well known that $n$-dimensional projective group gives rise to a non-homogenous representation of the Lie algebra $sl(n+1)$ on the polynomial functions of the projective space. Using Shen's mixed product for Witt algebras (also known as Larsson functor), we generalize the above representation of $sl(n+1)$ to a non-homogenous representation on the tensor space of any finite-dimensional irreducible $gl(n)$-module with the polynomial space. Moreover, the structure of such a representation is completely determined by employing projection operator techniques and well-known Kostant's characteristic identities for certain matrices with entries in the universal enveloping algebra. In particular, we obtain a new one parameter family of infinite-dimensional irreducible $sl(n+1)$-modules, which are in general not highest-weight type, for any given finite-dimensional irreducible $sl(n)$-module. The results could also be used to study the quantum field theory with the projective group as the symmetry.

math.RT↗

arXiv2009

Differential-Operator Representations of $S_n$ and Singular Vectors in Verma Modules

Given a weight of $sl(n,\mbb{C})$, we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator representation of the symmetric group $S_n$ on the related space of truncated power series. We prove that the solution space of the system of partial differential equations is exactly spanned by $\{\sgm(1)\mid \sgm\in S_n\}$. Moreover, the singular vectors of $sl(n,\mbb{C})$ in the Verma module are given by those $\sgm(1)$ that are polynomials. The well-known results of Verma, Bernstein-Gel'fand-Gel'fand and Jantzen for the case of $sl(n,\mbb{C})$ are naturally included in our almost elementary approach of partial differential equations.

math.RT↗

arXiv2009

Representations of Lie Algebras and Coding Theory

Linear codes with large minimal distances are important error correcting codes in information theory.Orthogonal codes have more applications in the other fields of mathematics. In this paper, we study the binary and ternary orthogonal codes generated by the weight matrices on finite-dimensional modules of simple Lie algebras. The Weyl groups of the Lie algebras act on these codes isometrically. It turns out that certain weight matrices of $sl(n,\mbb{C})$ and $o(2n,\mbb{C})$ generate doubly-even binary orthogonal codes and ternary orthogonal codes with large minimal distances. Moreover, we prove that the weight matrices of $F_4$, $E_6$, $E_7$ and $E_8$ on their minimal irreducible modules and adjoint modules all generate ternary orthogonal codes with large minimal distances. In determining the minimal distances, we have used the Weyl groups and branch rules of the irreducible representations of the related simple Lie algebras.

math.RT↗