低次对角交错调和多项式的基
Basis of Diagonally Alternating Harmonic Polynomials for low degree
摘要：
给定$n$单元格列表$L=[(p_1,q_1),...,(p_n, q_n)]$，其中$p_i, q_i\in \textbf{Z}_{\ge 0}$，我们设$Δ_L=\det |{(p_j!)^{-1}(q_j!)^{-1} x^{p_j}_iy^{q_j}_i} |$。对角交替多项式的空间由$\{Δ_L\}$构成，其中$L$在具有$n$单元的所有列表之间变化。对于$a>0$，算子$E_a=\sum_{i=1}^{n} y_i\partial_{x_i}^a$作用于对角交替多项式，Haiman证明了对角交替调和多项式的空间$A_n$被$\{E_λΔ_n\}$所张成。对于具有$t_m>...>t_1>0$的$t=(t_m,...,t_1)\in \textbf{Z}_{> 0}^m$，我们在这里考虑运算符$F_t=\det\big\|E_{t_{m-j+1}+(j-i)}\big\|$。我们的第一个结果是表明$F_tΔ_L$是$Δ_{L'}$的线性组合，其中$L'$是通过{ \sl移动} $\ell(t)=m$不同的细胞以某种确定的方式从$L$获得的。这允许我们控制$F_{t_{(1)}}... F_{t_{(r)}}Δ_n$形式的某些元素的前导项。我们用它来描述$A_n=\bigoplus A_n^{k,l}$的一些双齐性成分的显式基，其中$A_n^{k,l}=\hbox{Span}\{E_λΔ_n :\ell(λ)=l, |λ|=k\}$。更确切地说，我们给出了一个明确的基础$A_n^{k,l}$每当$k<n$。为此，我们介绍了一个新的变形Schensted插入一类特殊的舞台。这在分区和这个新的表格类之间产生了一个双射。这些表格$T$的组合学允许我们准确地知道$F_TΔ_n$的前导项，其中$F_T$是对应于$T$的列的运算符，并且每当$n$大于$T$的权重时。
英文摘要：
Given a list of $n$ cells $L=[(p_1,q_1),...,(p_n, q_n)]$ where $p_i, q_i\in \textbf{Z}_{\ge 0}$, we let $Δ_L=\det |{(p_j!)^{-1}(q_j!)^{-1} x^{p_j}_iy^{q_j}_i} |$. The space of diagonally alternating polynomials is spanned by $\{Δ_L\}$ where $L$ varies among all lists with $n$ cells. For $a>0$, the operators $E_a=\sum_{i=1}^{n} y_i\partial_{x_i}^a$ act on diagonally alternating polynomials and Haiman has shown that the space $A_n$ of diagonally alternating harmonic polynomials is spanned by $\{E_λΔ_n\}$. For $t=(t_m,...,t_1)\in \textbf{Z}_{> 0}^m$ with $t_m>...>t_1>0$, we consider here the operator $F_t=\det\big\|E_{t_{m-j+1}+(j-i)}\big\|$. Our first result is to show that $F_tΔ_L$ is a linear combination of $Δ_{L'}$ where $L'$ is obtained by {\sl moving} $\ell(t)=m$ distinct cells from $L$ in some determined fashion. This allows us to control the leading term of some elements of the form $F_{t_{(1)}}... F_{t_{(r)}}Δ_n$. We use this to describe explicit bases of some of the bihomogeneous components of $A_n=\bigoplus A_n^{k,l}$ where $A_n^{k,l}=\hbox{Span}\{E_λΔ_n :\ell(λ)=l, |λ|=k\}$. More precisely we give an explicit basis of $A_n^{k,l}$ whenever $k<n$. To this end, we introduce a new variation of Schensted insertion on a special class of tableaux. This produces a bijection between partitions and this new class of tableaux. The combinatorics of those tableaux $T$ allows us to know exactly the leading term of $F_TΔ_n$ where $F_T$ is the operator corresponding to the columns of $T$ and whenever $n$ is bigger than the weight of $T$.
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