具有反射对称性的交错符号矩阵与平面分拆：$n+3$ 对等价统计量与一个柯西型恒等式
Alternating sign matrices with reflective symmetry and plane partitions: $n+3$ pairs of equivalent statistics and a Cauchy-type identity
AI总结：
本文推广了垂直对称交错符号矩阵与带洞六边形环面平铺的等数性，引入 $n+3$ 个参数并证明多变量生成函数一致，还给出辛特征展开，为构造双射和柯西型恒等式提供关键步骤。
摘要：
已知 $2n+1$ 阶垂直对称交错符号矩阵（VSASMs）与边长为 $2n+2$, $2n$, $2n+2$, $2n$, $2n+2$, $2n$ 且具有大小为 $2$ 的中心三角形洞的六边形上的环面平铺（lozenge tilings）在数量上相等，且这些平铺同时具有循环对称性和垂直对称性，但找到一个明确的双射来证明这一点属于双射组合学中最困难的问题之一。为了构造这样的双射，我们通过为这两类对象引入某些自然扩展以及 $n+3$ 个参数来推广这一结果，并证明关于这些参数的多变量生成函数是一致的。这是从常数个等分布统计量到 $n$ 个线性数量统计量的重要一步。VSASMs 与环面平铺的数量相等性是该结果的一个简单推论，通过将生成函数特化为这两类对象的带符号枚举，然后应用某些符号反转对合（sign-reversing involutions）即可得到。另一个主要结果涉及将多变量生成函数展开为辛特征（symplectic characters），其展开式是对全对称自互补平面分拆（totally symmetric self-complementary plane partitions）求和，这与普通交错符号矩阵（ASMs）中 Schur 展开可写成对全对称平面分拆求和的情形完全类似。这令人兴奋，因为它让人联想到著名的柯西恒等式（Cauchy identity），而柯西恒等式确实有使用 Robinson-Schensted-Knuth 对应的双射证明，因此这一结果提出了一个问题：是否存在 Robinson-Schensted-Knuth 对应的变体，最终能导致一个双射证明。
英文摘要：
Vertically symmetric alternating sign matrices (VSASMs) of order $2n+1$ are known to be equinumerous with lozenge tilings of a hexagon with side lengths $2n+2$, $2n$, $2n+2$, $2n$, $2n+2$, $2n$ and a central triangular hole of size $2$ that exhibit a cyclical as well as a vertical symmetry, but finding an explicit bijection proving this belongs to the most difficult problems in bijective combinatorics. Towards constructing such a bijection, we generalize the result by introducing certain natural extensions for both objects along with $n+3$ parameters and show that the multivariate generating functions with respect to these parameters coincide. This is a significant step from a constant number of equidistributed statistics to a linear number of statistics in $n$. The equinumeracy of VSASMs and the lozenge tilings is then an easy consequence of this result, which is obtained by specializing the generating functions to signed enumerations for both types of objects and then applying certain sign-reversing involutions. Another main result concerns the expansion of the multivariate generating function into symplectic characters as a sum over totally symmetric self-complementary plane partitions, which is in perfect analogy to the situation for ordinary ASMs where the Schur expansion can be written as a sum over totally symmetric plane partitions. This is exciting as it is reminiscent of the well-known Cauchy identity, and the Cauchy identity does have a bijective proof using the Robinson-Schensted-Knuth correspondence, and thus the result raises the question of whether there is a variation of the Robinson-Schensted-Knuth correspondence that does eventually lead to a bijective proof.
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