Bases for diagonally alternating harmonic polynomials of low degree

Given a list of n cells L = [ ( p 1 , q 1 ) , … , ( p n , q n ) ] where p i , q i ∈ Z ⩾ 0 , we let Δ L = det ‖ ( p j ! ) − 1 ( q j ! ) − 1 x i p j y i q j ‖ . 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 = ∑ i = 1 n y i ∂ x i a act on diagonally alternating polynomials. Haiman has shown that the space A n of diagonally alternating harmonic polynomials is spanned by { E λ Δ n } where λ = ( λ 1 , … , λ ℓ ) varies among all partitions, E λ = E λ 1 ⋯ E λ ℓ and Δ n = det ‖ ( ( n − j ) ! ) − 1 x i n − j ‖ . For t = ( t m , … , t 1 ) ∈ Z > 0 m with t m > ⋯ > t 1 > 0 , we consider here the operator F t = det ‖ E t m − j + 1 + ( j − i ) ‖ . Our first result is to show that F t Δ L is a linear combination of Δ L ′ 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 F t ( 1 ) ⋯ F t ( r ) Δ n . We use this to describe explicit bases of some of the bihomogeneous components of A n = ⊕ A n k , l where A n k , l = Span { E λ Δ n : ℓ ( 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 these tableaux T allow us to know exactly the leading term of F T Δ n where F T is the operator corresponding to the columns of T, whenever n is greater than the weight of T.