https://hal.inria.fr/hal-01185424Assaf, SamiSamiAssafDepartment of Mathematics [MIT] - MIT - Massachusetts Institute of TechnologyGarsia, AdrianoAdrianoGarsiaDepartment of Mathematics [MIT] - MIT - Massachusetts Institute of TechnologyA kicking basis for the two column Garsia-Haiman modulesHAL CCSD2009Macdonald polynomialsGarsia-Haiman modulescombinatorial basis[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO][INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Episciences Iam, CoordinationKrattenthaler, Christian and Strehl, Volker and Kauers, Manuel2015-08-20 11:09:042019-06-26 14:48:032015-08-24 10:04:01enConference papershttps://hal.inria.fr/hal-01185424/document10.46298/dmtcs.2732application/pdf1In 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.