In a recent paper, Ayyer and Behrend present for a wide class of partitions factorizations of Schur polynomials with an even number of variables where half of the variables are the reciprocals of the others into symplectic and/or orthogonal group characters, thereby generalizing results of Ciucu and Krattenthaler for rectangular shapes. Their proofs proceed by manipulations of determinants underlying the characters. The purpose of the current paper is to provide bijective proofs of such factorizations. The quantities involved have known combinatorial interpretations in terms of Gelfand-Tsetlin patterns of various types or half Gelfand-Tsetlin patterns, which can in turn be transformed into perfect matchings of weighted trapezoidal honeycomb graphs. An important ingredient is then Ciucu's theorem for graphs with reflective symmetry. However, before being able to apply it, we need to employ a certain averaging procedure in order to achieve symmetric edge weights. This procedure is based on a “randomized” bijection, which can however also be turned into a classical bijection. For one type of Schur polynomial factorization, we also need an additional graph operation that almost doubles the underlying graph. Finally, our combinatorial proofs reveal that the factorizations under consideration can in fact also be generalized to skew shapes as discussed at the end of the paper.

在最近的一篇论文中，Ayyer和Behrend提出了具有偶数个变量的Schur多项式的各种分区分解，其中一半变量是其他变量倒数为辛和/或正交的组字符，从而推广了Ciucu的结果和矩形的Krattenthaler。他们的证明是通过操纵字符背后的行列式进行的。本文的目的是提供这种分解的双射证明。涉及的数量在各种类型的Gelfand-Tsetlin模式或半Gelfand-Tsetlin模式方面都具有已知的组合解释，这些反过来又可以转换成加权梯形蜂窝图的完美匹配。然后，重要的组成部分是具有反射对称性的图的Ciucu定理。然而，在能够应用它之前，我们需要采用一定的平均过程以实现对称的边缘权重。此过程基于“随机”双射，但是也可以将其转换为经典双射。对于一种类型的Schur多项式因式分解，我们还需要额外的图操作，该操作几乎会使基础图加倍。最后，我们的组合证明表明，所考虑的因式分解实际上也可以概括为偏斜形状，如本文结尾处所述。我们还需要执行其他图形操作，从而使基础图形几乎翻倍。最后，我们的组合证明表明，所考虑的因式分解实际上也可以概括为偏斜形状，如本文结尾处所述。我们还需要执行其他图形操作，从而使基础图形几乎翻倍。最后，我们的组合证明表明，所考虑的因式分解实际上也可以概括为偏斜形状，如本文结尾处所述。