Constant Term Methods in the Theory of Tesler Matrices and Macdonald Polynomial Operators

The Tesler matrices with hook sums ( a 1 , a 2 , . . . , a n ) are non-negative integral upper triangular matrices, whose i th diagonal element plus the sum of the entries in the arm of its (french) hook minus the sum of the entries in its leg is equal to a i for all i . In a recent paper [ 6 ], the second author expressed the Hilbert series of the Diagonal Harmonic modules as a weighted sum of the family of Tesler matrices with hook weights (1, 1, . . . , 1). In this paper we use the constant term algorithm developed by the third author to obtain a Macdonald polynomial interpretation of these weighted sum of Tesler matrices for arbitrary hook weights. In particular, we also obtain new and illuminating proofs of the results in [ 6 ].