Basis of Diagonally Alternating Harmonic Polynomials for low degree

Details Basis of Diagonally Alternating Harmonic Polynomials for low degree Basis of Diagonally Alternating Harmonic Polynomials for low degree

2009-05-04

arxiv.org/abs/0905.0377v2

JCT-A 118 (2011) 37--57

Mathematics

Combinatorics

To appear in JCT-A; 21 pages, one PDF figure

Scientific paper

10.1016/j.jcta.2010.04.002

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

Affiliated with

Also associated with

No associations

Rating

Basis of Diagonally Alternating Harmonic Polynomials for low degree does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Basis of Diagonally Alternating Harmonic Polynomials for low degree, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Basis of Diagonally Alternating Harmonic Polynomials for low degree will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-199528