欧拉插入算子与皮耶里规则的欧拉形式
Eulerian insertion operators and an Eulerian form of the Pieri rule
- School of Mathematics and Statistics, Shandong University of Technology（山东理工大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究通过对称函数理论统一解释普通与主指标插入算子，推导欧拉插入算子表达式，建立对称函数环到插入算子代数的同态，得到皮耶里规则的欧拉形式。
AI中文摘要：
我们研究了向多重集排列中插入若干个新的最大字母副本所得到的算子。设$G_r$表示插入$r$个新的最大字母副本的算子。经过变量替换$δ=y-x$、$u=x/y$以及$E=u\theta_u$，我们得到$$G_r=\frac{δ^r}{r!}E(E+1)\ncdots(E+r-1).$$其生成级数通过有理代换发挥作用，由此可推导出合成法则。我们的主要结果为普通插入算子和主指标算子提供了统一的对称函数解释。对于$N\neq0$，定义$Φ_N(F_{N,S})=x^{|S|+1}y^{N-|S|}$。我们证明，与完全齐次对称函数$h_r$相乘对应着普通插入算子：$Φ_{N+r}(h_r f)=G_rΦ_N(f)$，其中$f∈\nQSym_N$。主指标也存在对应的平行特化。一种反向有限主特化将与$h_r$相乘的运算映射为算子$Q_r$，它是$q$位移$Θ_qf(t)=f(qt)$的多项式。因此，普通插入算子和主指标算子都源自同一个乘法算子$f↦h_r f$。由于函数$h_r$自由生成对称函数环，映射$h_r↦G_r$可扩展为一个代数同态。我们确定了该同态的核以及每个齐次分量的像。Schur函数的像满足Littlewood--Richardson乘法恒等式，且单行情形给出了皮耶里规则的欧拉形式。
英文摘要：
We study the operators obtained by inserting copies of a new largest letter into multiset permutations. Let $G_r$ denote the operator which inserts $r$ copies of a new largest letter. After the change of variables $δ=y-x$, $u=x/y$, and $E=u\partial_u$, we find that $$G_r=\frac{δ^r}{r!}E(E+1)\cdots(E+r-1).$$ Its generating series acts by a rational substitution, which yields the composition law. Our main result gives a common symmetric-function explanation for the ordinary and major-index operators. For $N\geq 0$, define $Φ_N(F_{N,S})=x^{|S|+1}y^{N-|S|}$. We prove that multiplication by the complete homogeneous symmetric function $h_r$ becomes the ordinary insertion operator: $Φ_{N+r}(h_r f)=G_rΦ_N(f)$, where $f\in\mathrm{QSym}_N$. There is a parallel specialization for the major index. A reverse finite principal specialization sends multiplication by $h_r$ to an operator $Q_r$, which is a polynomial in the $q$-shift $Θ_qf(t)=f(qt)$. Thus the ordinary and major-index operators arise from the same multiplication operator $f\mapsto h_r f$. Since the functions $h_r$ freely generate the ring of symmetric functions, the assignment $h_r\mapsto G_r$ extends to an algebra homomorphism. We determine the kernel of this homomorphism and the image of every homogeneous component. The images of Schur functions satisfy the Littlewood--Richardson multiplication identities, and the one-row case gives an Eulerian form of the Pieri rule.
