A proof of the \(n!\) conjecture for generalized hooks
\textit{A. M. Garsia} and \textit{M. Haiman} [Electron. J. Comb. 3, No. 2, Paper R24, 60 p. (1996)] posed the following conjecture about Macdonald polynomials \(H_\mu(x; q, t)\): for a partition \(\mu\) of \(n\), a determinant polynomial \(\Delta_\mu(X_n, Y_n)\) has \(n!\) linearly independent partial derivatives, i.e. the dimension of the vector space of these derivatives is \(n!\). The author proves this conjecture, using some previous results of the above-mentioned authors and himself.
scientific article; zbMATH DE number 909610