论文标题
交替的符号矩阵和完全对称的平面分区
Alternating sign matrices and totally symmetric plane partitions
论文作者
论文摘要
我们研究了与交替符号矩阵相对于其反转数，互补反转数和唯一$ 1 $在上排行中的位置的对称多项式的Schur多项式扩展。我们证明，扩展可以表示为完全对称平面分区的总和，并且我们还能够确定系数。这在交替的符号矩阵和一类平面分区之间建立了新的联系，从而补充了交替的符号矩阵与完全对称的自我平面平面分区以及下降平面分区的事实。作为副产品，我们从完全对称的平面分区获得了一个有趣的地图到Dyck路径。该证明是基于一种新的，相当一般的反对称器到确定的公式。
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.
