Alternating sign matrices and totally symmetric plane partitions
Alternating sign matrices and totally symmetric plane partitions
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交替符号矩阵和完全对称平面划分
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
V. Tewari
中科院分区：
文献类型：
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作者：
F. Aigner;Ilse Fischer;M. Konvalinka;Philippe Nadeau;V. Tewari
本文研究了与交替符号矩阵的精细化枚举有关的对称多项式族的舒尔多项式展开式，讨论了它们的反转数、互补反转数和唯一$1$在上行的位置。我们证明了展开式可以表示为完全对称平面分区的和，并且我们也能够确定系数。这在交替符号矩阵与一类平面划分之间建立了新的联系，从而补充了交替符号矩阵与完全对称自互补平面划分以及与下降平面划分相等的事实。作为一个副产品，我们得到了一个有趣的映射，从完全对称平面分区到Dyck路径。这个证明是基于一个新的，相当一般的反对称-行列式公式。
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.