Linearly Independent Products of Rectangularly Complementary Schur Functions

Schur's theorem in integer lattices

A standard proof of Schur's Theorem yields that any $r$-coloring of $\{1,2,\dots,R_r-1\}$ yields a monochromatic solution to $x+y=z$, where $R_r$ is the classical $r$-color Ramsey number, the minimum $N$ such that any $r$-coloring of a complete graph on $N$ vertices yields a monochromatic triangle. We explore generalizations and modifications of this result in higher dimensional integer lattices, showing in particular that if $k\geq d+1$, then any $r$-coloring of $\{1,2,\dots,R_r(k)^d-1\}^d$ yields a monochromatic solution to $x_1+\cdots+x_{k-1}=x_k$ with $\{x_1,\dots,x_d\}$ linearly independe…

Schur乘子在$q$-永久函数的Pólya转换问题中的极限

The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function

本文研究$q$-永久函数的广义Pólya转换问题，建立了从低维代数灵活性到高维组合刚性的尖锐阈值，并分类了Schur乘子保持子空间。

Linear transformations of vertex operator presentations of Hall-Littlewood polynomials

We study the effect of linear transformations on quantum fields with applications to vertex operator presentations of symmetric functions. Properties of linearly transformed quantum fields and corresponding transformations of Hall-Littlewood polynomials are described, including preservation of commutation relations, stability, explicit combinatorial formulas and generating functions. We prove that specializations of linearly transformed Hall-Littlewood polynomials describe all polynomial tau functions of the KP and the BKP hierarchy. Examples of linear transformations are related to multiparam…

克莱伯猜想与对称函数的互补积

Kleber's conjecture and complementary products of symmetric functions

研究任意交换环\(R\)上舒尔函数的克莱伯矩形补猜想，通过分量分裂的一般独立性定理证明相关乘积线性无关，还得出小池 - 寺田通用特征积线性无关，以及特征零域上单项式对称函数类似结果和\(\mathbb{Z}\)上整线性无关性。