论文标题
dynkin类型$ \ mathbb {a} _n $非负posets的Coxeter类型分类
A Coxeter type classification of Dynkin type $\mathbb{A}_n$ non-negative posets
论文作者
论文摘要
我们继续进行有限连接的POSET $ i $的Coxeter光谱分析，从某种意义上说，它们的对称gram矩阵$ g_i：= \ frac {1} {2} {2} {2}（c_i + c_i + c_i^{tr} {tr} \ in \ mathbb {m} _ {m} _ {m} _ {m} $ n \ geq 0 $，其中$ c_i \ in \ mathbb {m} _m（\ mathbb {z}）$是$ i $编码关系$ \ preceq_i $的$ i $的发射矩阵。我们扩展了[fundam的结果。 Inform。，139.4（2015），347--367]，并给出有限连接的完整的Coxeter光谱分类Posets $ i $ i $ i $ i $ i $ type $ \ mathbb {a} _n $。我们表明，具有$ | i |> 1 $的POSETS $ i $，恰好收益$ \ lfloor \ frac {m} {2} {2} \ rfloor $ coxeter类型，其中一种描述了正（即$ n = m $）。我们给出确切的描述并计算每种类型的POSET数量。此外，我们证明，给定一对此类posets $ i $和$ j $，发射率$ c_i $和$ c_j $是$ \ mathbb {z} $ - 仅当$ \ mathbf {specc} _i _i _i = = \ mathbf {specc} _j {specc} _j _j $ and conders and conculation an $ \ mathbb {z} $ - 可逆矩阵定义了这样的$ \ mathbb {z} $ - 在多项式时间内的一致性。
We continue the Coxeter spectral analysis of finite connected posets $I$ that are non-negative in the sense that their symmetric Gram matrix $G_I:=\frac{1}{2}(C_I + C_I^{tr})\in\mathbb{M}_{m}(\mathbb{Q})$ is positive semi-definite of rank $n\geq 0$, where $C_I\in\mathbb{M}_m(\mathbb{Z})$ is the incidence matrix of $I$ encoding the relation $\preceq_I$. We extend the results of [Fundam. Inform., 139.4(2015), 347--367] and give a complete Coxeter spectral classification of finite connected posets $I$ of Dynkin type $\mathbb{A}_n$. We show that such posets $I$, with $|I|>1$, yield exactly $\lfloor\frac{m}{2}\rfloor$ Coxeter types, one of which describes the positive (i.e., with $n=m$) ones. We give an exact description and calculate the number of posets of every type. Moreover, we prove that, given a pair of such posets $I$ and $J$, the incidence matrices $C_I$ and $C_J$ are $\mathbb{Z}$-congruent if and only if $\mathbf{specc}_I = \mathbf{specc}_J$, and present deterministic algorithms that calculate a $\mathbb{Z}$-invertible matrix defining such a $\mathbb{Z}$-congruence in a polynomial time.
