An analogue of Schur functions for the plane partitions

Abstract

An attempt is described to extend the notion of Schur functions from Young diagrams to plane partitions. The suggestion is to use the recursion in the partition size, which is easily generalized and deformed. This opens a possibility to obtain Macdonald polynomials by a change of recursion coefficients and taking appropriate limit from three to two dimensions -- though details still remain to be worked out. Another perspective is opened by the observation of a rich non-abelian structure, extending that of commuting cut-and-join operators, for which the discovered 3-Schurs are the common eigenfunctions.

arXiv comment12 pages

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• Hunt for 3-Schur polynomials
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• On generalized Macdonald polynomials
• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• MacMahon KZ equation for Ding-Iohara-Miki algebra
• Equating Schur Functions
• Weyl Mutations in Quiver Yangians
• Quiver matrix model of ADHM type and BPS state counting in diverse dimensions