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arXiv · 2005.12448

Alternating sign matrices and totally symmetric plane partitions

Florian Aigner·Ilse Fischer·Matjaž Konvalinka·Philippe Nadeau·Vasu Tewari
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Abstract

We study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices with respect to their inversion number, complementary inversion number and the position of the unique $1$ in the top row. We prove that the expansion can be expressed as a sum over totally symmetric plane partitions and we are also able to determine the coefficients. This establishes a new connection between alternating sign matrices and a class of plane partitions, thereby complementing the fact that alternating sign matrices are equinumerous with totally symmetric self-complementary plane partitions as well as with descending plane partitions. As a by-product we obtain an interesting map from totally symmetric plane partitions to Dyck paths. The proof is based on a new, quite general antisymmetrizer-to-determinant formula.

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Florian Aigner, Ilse Fischer, Matjaž Konvalinka, Philippe Nadeau, Vasu Tewari. 2020-05-26. Alternating sign matrices and totally symmetric plane partitions. https://arxiv.org/abs/2005.12448

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