On certain spaces of lattice diagram polynomials

Jean-Christophe Aval (1, 2)

1 A2X - Théorie des Nombres et Algorithmique Arithmétique
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

The aim of this work is to study some lattice diagram determinants $\Delta_L(X,Y)$. We recall that $M_L$ denotes the space of all partial derivatives of $\Delta_L$. In this paper, we want to study the space $M^k_{i,j}(X,Y)$ which is defined as the sum of $M_L$ spaces where the lattice diagrams $L$ are obtained by removing $k$ cells from a given partition, these cells being in the ``shadow'' of a given cell $(i,j)$ in a fixed Ferrers diagram. We obtain an upper bound for the dimension of the resulting space $M^k_{i,j}(X,Y)$, that we conjecture to be optimal. This dimension is a multiple of $n!$ and thus we obtain a generalization of the $n!$ conjecture. Moreover, these upper bounds associated to nice properties of some special symmetric differential operators (the ``shift'' operators) allow us to construct explicit bases in the case of one set of variables, i.e. for the subspace $M^k_{i,j}(X)$ consisting of elements of $0$ $Y$-degree.

Domaines: Combinatoire [math.CO]

Jean-Christophe Aval. On certain spaces of lattice diagram polynomials. Discrete Mathematics, 2002, 256, pp.557-575. ⟨hal-00185525⟩