A kicking basis for the two column Garsia-Haiman modules
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009)
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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.
@article{DMTCS_2009_special_256_a54,
author = {Assaf, Sami and Garsia, Adriano},
title = {A kicking basis for the two column {Garsia-Haiman} modules},
journal = {Discrete mathematics & theoretical computer science},
year = {2009},
volume = {DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)},
doi = {10.46298/dmtcs.2732},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2732/}
}
TY - JOUR AU - Assaf, Sami AU - Garsia, Adriano TI - A kicking basis for the two column Garsia-Haiman modules JO - Discrete mathematics & theoretical computer science PY - 2009 VL - DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) UR - http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2732/ DO - 10.46298/dmtcs.2732 LA - en ID - DMTCS_2009_special_256_a54 ER -
%0 Journal Article %A Assaf, Sami %A Garsia, Adriano %T A kicking basis for the two column Garsia-Haiman modules %J Discrete mathematics & theoretical computer science %D 2009 %V DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) %U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2732/ %R 10.46298/dmtcs.2732 %G en %F DMTCS_2009_special_256_a54
Assaf, Sami; Garsia, Adriano. A kicking basis for the two column Garsia-Haiman modules. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009). doi: 10.46298/dmtcs.2732
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