Basis of Diagonally Alternating Harmonic Polynomials for low degree

Abstract

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 $\Delta_L=\det |{(p_j!)^{-1}(q_j!)^{-1} x^{p_j}_iy^{q_j}_i} |$. The space of diagonally alternating polynomials is spanned by $\{\Delta_L\}$ 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 $\{E_\lambda\Delta_n\}$. 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 $F_t\Delta_L$ is a linear combination of $\Delta_{L'}$ 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 $F_{t_{(1)}}... F_{t_{(r)}}\Delta_n$. We use this to describe explicit bases of some of the bihomogeneous components of $A_n=\bigoplus A_n^{k,l}$ where $A_n^{k,l}=\hbox{Span}\{E_\lambda\Delta_n :\ell(\lambda)=l, |\lambda|=k\}$. 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 $F_T\Delta_n$ where $F_T$ is the operator corresponding to the columns of $T$ and whenever $n$ is bigger than the weight of $T$.