论文标题
dynkin型$ \ mathbb {a} _n $的非负posets的结构
Structure of non-negative posets of Dynkin type $\mathbb{A}_n$
论文作者
论文摘要
如果对称gram matrix $ g_i：= \ frac {1} {2} {2} {2} {2} {2} {2}（c_i + c_i + c_i + c_i + c_i^{tr} {tr}）\ in \ in \ mathbb {mathbb {mathbb {m mathbb {m} r，则p poset $ i =（\ {1，\ {1，\ ldots，\ ldots，n \}，\ leq_i）$被称为非阴性。 semi-definite，其中$ c_i \ in \ mathbb {m} _n（\ mathbb {z}）$是$（0,1）$ - 编码关系$ \ leq_i $的$（0,1）$ - 矩阵。每个这样的连接的poset $ i $，to $ \ mathbb {z} $ - $ g_i $矩阵的一致性，由唯一简单的dynkin图$ \ mathrm {dyn andrm {dyn} _i \ in \ in \ { \ mathbb {d} _m，\ m athbb {e} _6，\ mathbb {e} _7，\ mathbb {e} _8 \} $。我们表明$ \ mathrm {dyn} _i = \ mathbb {a} _n $意味着矩阵$ g_i $是等级$ n $或$ n-1 $。此外，我们描绘了Hasse Digraphs $ \ Mathcal {h}（i）的明确形状，所有此类posets〜 $ i $的$并为其编号设计了公式。
A poset $I=(\{1,\ldots, n\}, \leq_I)$ is called non-negative if the symmetric Gram matrix $G_I:=\frac{1}{2}(C_I + C_I^{tr})\in\mathbb{M}_n(\mathbb{R})$ is positive semi-definite, where $C_I\in\mathbb{M}_n(\mathbb{Z})$ is the $(0,1)$-matrix encoding the relation $\leq_I$. Every such a connected poset $I$, up to the $\mathbb{Z}$-congruence of the $G_I$ matrix, is determined by a unique simply-laced Dynkin diagram $\mathrm{Dyn}_I\in\{\mathbb{A}_m, \mathbb{D}_m,\mathbb{E}_6,\mathbb{E}_7,\mathbb{E}_8\}$. We show that $\mathrm{Dyn}_I=\mathbb{A}_n$ implies that the matrix $G_I$ is of rank $n$ or $n-1$. Moreover, we depict explicit shapes of Hasse digraphs $\mathcal{H}(I)$ of all such posets~$I$ and devise formulae for their number.
