Bases for diagonally alternating harmonic polynomials of low degree

AbstractGiven a list of n cells L=[(p1,q1),…,(pn,qn)] where pi,qi∈Z⩾0, we let ΔL=det‖(pj!)−1(qj!)−1xipjyiqj‖. The space of diagonally alternating polynomials is spanned by {ΔL} where L varies among all lists with n cells. For a>0, the operators Ea=∑i=1nyi∂xia act on diagonally alternating polynomials. Haiman has shown that the space An of diagonally alternating harmonic polynomials is spanned by {EλΔn} where λ=(λ1,…,λℓ) varies among all partitions, Eλ=Eλ1⋯Eλℓ and Δn=det‖((n−j)!)−1xin−j‖. For t=(tm,…,t1)∈Z>0m with tm>⋯>t1>0, we consider here the operator Ft=det‖Etm−j+1+(j−i)‖. Our first result is to show that FtΔL is a linear combination of ΔL′ where L′ is obtained by moving ℓ(t)=m distinct cells of L in some determined fashion. This allows us to control the leading term of some elements of the form Ft(1)⋯Ft(r)Δn. We use this to describe explicit bases of some of the bihomogeneous components of An=⊕Ank,l where Ank,l=Span{EλΔn:ℓ(λ)=l,|λ|=k}. More precisely, we give an explicit basis of Ank,l whenever k