Schur Partial Derivative Operators

Details Schur Partial Derivative Operators Schur Partial Derivative Operators

2001-11-22

arxiv.org/abs/math/0111246v1

European J. Combin. 26 (2005), no. 6, 785--794.

Mathematics

Combinatorics

8 pages, LaTeX

Scientific paper

A lattice diagram is a finite list L=((p_1,q_1),...,(p_n,q_n) of lattice cells. The corresponding lattice diagram determinant is \Delta_L(X;Y)=\det \| x_i^{p_j}y_i^{q_j} \|. These lattice diagram determinants are crucial in the study of the so-called ``n! conjecture'' of A. Garsia and M. Haiman. The space M_L is the space spanned by all partial derivatives of \Delta_L(X;Y). The ``shift operators'', which are particular partial symmetric derivative operators are very useful in the comprehension of the structure of the M_L spaces. We describe here how a Schur function partial derivative operator acts on lattice diagrams with distinct cells in the positive quadrant.

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