AI 中文总结
该研究刻画右钥匙表，发展m对称函数环的舒尔函数在t=0处的组合理论，证明相关柯西恒等式，关联m对称舒尔函数与几乎对称舒尔函数并推导行列式公式。
AI 中文摘要
m对称函数环$R_m$由形式幂级数构成，这类幂级数在变量$x_{m+1},x_{m+2},\dots$上对称，但在前m个变量上无对称性。我们为$R_m$的舒尔函数在t=0处的特化发展了一套组合理论。我们的主要工具是右钥匙表的新刻画，将其定义为T的子表的读单词的递减子词集合的上确界。该刻画在初等Knuth变换下不变，因此与RSK对应兼容。我们通过这种方式得到了t=0处m对称舒尔函数的半标准表生成函数，同时给出了$R_m$中柯西恒等式的组合证明。随后，m对称舒尔函数及其对偶分别被等同于Demazure原子和Demazure特征。限制到最后m个变量，我们的对应通过普通RSK给出了Lascoux关于Demazure特征和原子的非对称柯西恒等式的证明。作为进一步应用，我们通过单三角基变换矩阵将t=0处的m对称舒尔函数与几乎对称舒尔函数关联起来，得到了这两类函数的表生成函数和柯西恒等式，并推导出三种不同基的雅可比-特里德里型行列式公式。
英文摘要
The ring $R_m$ of $m$-symmetric functions consists of the formal power series that are symmetric in the variables $x_{m+1},x_{m+2},\dots$ but carry no symmetry in the first $m$ variables. We develop a combinatorial theory for the specialization at $t=0$ of the Schur functions of $R_m$. Our main tool is a new characterization of right key tableaux as suprema of the sets of decreasing subwords of the reading words of the subtableaux of $T$. Being invariant under elementary Knuth transformations, this characterization is compatible with the RSK correspondence. We obtain in this way a generating function over semistandard tableaux for the $m$-symmetric Schur functions at $t=0$, together with a combinatorial proof of a Cauchy identity in $R_m$. The $m$-symmetric Schur functions and their dual are then respectively identified with Demazure atoms and Demazure characters. Restricted to the last $m$ variables, our correspondence specializes to a proof, by ordinary RSK, of Lascoux's nonsymmetric Cauchy identity for Demazure characters and atoms. As further applications, we relate the $m$-symmetric Schur functions at $t=0$ to the almost symmetric Schur functions through a unitriangular change-of-basis matrix, obtain tableau generating functions and Cauchy identities for both families, and derive Jacobi-Trudi type determinantal formulas for three different bases.
Comments32 pages
