半环上的平面流与Plücker型二次关系
Planar flows and Plücker's type quadratic relations over semirings
摘要：
众所周知，由于Lindström，（实或复）矩阵的子矩阵可以用平面有向图中流的权重来表示。另一个经典的事实是，有大量的齐次二次关系涉及旗子式，或相应旗流形的普吕克坐标。推广和统一这些事实和他们的热带同行，我们认为广泛的一类功能$2^{[n]}$所产生的流在一个可平面图，并采取值在任意交换半环，其中$[n]=\{1,2,\ldots,n\}$。我们表明，“普适的”齐次二次关系满足这样的功能，可以描述在某些匹配，并作为一个结果，给出组合的充要条件的子集的$[n]$确定这些关系。
英文摘要：
It is well known, due to Lindström, that the minors of a (real or complex) matrix can be expressed in terms of weights of flows in a planar directed graph. Another classical fact is that there are plenty of homogeneous quadratic relations involving flag minors, or Plücker coordinates of the corresponding flag manifold. Generalizing and unifying these facts and their tropical counterparts, we consider a wide class of functions on $2^{[n]}$ that are generated by flows in a planar graph and take values in an arbitrary commutative semiring, where $[n]=\{1,2,\ldots,n\}$. We show that the ``universal'' homogeneous quadratic relations fulfilled by such functions can be described in terms of certain matchings, and as a consequence, give combinatorial necessary and sufficient conditions on the collections of subsets of $[n]$ determining these relations.
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