It was Adriano Garsia's idea to construct an appropriate module in order to prove positivity (as was done in his previous joint work with Procesi on Schur positivity of Kostka–Foulkes polynomials).
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In fact, we can obtain in this manner the Schur functions, the Hall–Littlewood symmetric functions, the Jack symmetric functions, the zonal symmetric functions, the zonal spherical functions, and the elementary and monomial symmetric functions.
Goldbach's conjecture | ''n''! conjecture | n! conjecture | Kato's conjecture | Calabi conjecture | Weil conjecture | ''Uncle Petros and Goldbach's Conjecture'' by Apostolos Doxiadis | Uncle Petros and Goldbach's Conjecture | Schanuel's conjecture | Pollock's conjecture | Mumford conjecture | Kepler conjecture | Heawood conjecture | Chang's conjecture | Catalan's conjecture | Blattner's conjecture | Beal's conjecture |
These conjectures are now proved; the hardest and final step was proving the positivity, which was done by Mark Haiman (2001), by proving the n! conjecture.
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The n! conjecture of Adriano Garsia and Mark Haiman states that for each partition μ of n the space