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A kicking basis for the two column Garsia-Haiman modules

Abstract : In the early 1990s, Garsia and Haiman conjectured that the dimension of the Garsia-Haiman module $R_{\mu}$ is $n!$, and they showed that the resolution of this conjecture implies the Macdonald Positivity Conjecture. Haiman proved these conjectures in 2001 using algebraic geometry, but the question remains to find an explicit basis for $R_{\mu}$ which would give a simple proof of the dimension. Using the theory of Orbit Harmonics developed by Garsia and Haiman, we present a "kicking basis" for $R_{\mu}$ when $\mu$ has two columns.
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Sami Assaf, Adriano Garsia. A kicking basis for the two column Garsia-Haiman modules. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.103-114, ⟨10.46298/dmtcs.2732⟩. ⟨hal-01185424⟩



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