论文标题
正交成型在部分场上的可表示
Representability of orthogonal matroids over partial fields
论文作者
论文摘要
令$ r \ leqslant n $为非负整数，让$ n = \ binom {n} {r} - 1 $。对于有限套件$ e = [n] $和semple意义上的部分字段$ e = [n] $的矩阵$ m $ r $的$ r $ - 众所周知，以下是等效的：（a）$ m $在$ k $上代表； （b）有一个$ p =（p_j）\ in {\ bf p}^n（k）$带有支持$ m $（这意味着$ \ \ text {supp}（p）：= \ {j \ in \ binom {e} Grassmann-Plücker方程； （c）在{\ bf p}^n（k）$中有一个点$ p =（p_j）\，只需支撑$ m $就可以满足3-期Grassmann-Plücker方程。此外，根据纳尔逊P. Nelson的定理，几乎所有的矩形（含义渐近100％）在任何部分场上都不可表示。从Gelfand-Serganova的意义上讲，我们证明了这些事实的类似物，这些事实是Gelfand-Serganova的意义，在Bouchet的意义上，它们甚至相当于Delta-Matroids。
Let $r \leqslant n$ be nonnegative integers, and let $N = \binom{n}{r} - 1$. For a matroid $M$ of rank $r$ on the finite set $E = [n]$ and a partial field $k$ in the sense of Semple--Whittle, it is known that the following are equivalent: (a) $M$ is representable over $k$; (b) there is a point $p = (p_J) \in {\bf P}^N(k)$ with support $M$ (meaning that $\text{Supp}(p) := \{J \in \binom{E}{r} \; \vert \; p_J \ne 0\}$ of $p$ is the set of bases of $M$) satisfying the Grassmann-Plücker equations; and (c) there is a point $p = (p_J) \in {\bf P}^N(k)$ with support $M$ satisfying just the 3-term Grassmann-Plücker equations. Moreover, by a theorem of P. Nelson, almost all matroids (meaning asymptotically 100%) are not representable over any partial field. We prove analogues of these facts for Lagrangian orthogonal matroids in the sense of Gelfand-Serganova, which are equivalent to even Delta-matroids in the sense of Bouchet.
