论文标题
皮带多面体和渗透螺旋体的投影和角度总和
Projections and angle sums of belt polytopes and permutohedra
论文作者
论文摘要
令$ p \ subset \ mathbb r^n $为皮带polytope，那就是多层型，其正常风扇与某些超平面布置$ \ Mathcal a $相吻合。另外，令$ g：\ mathbb r^n \ to \ mathbb r^d $为完整等级的线性地图，其内核相对于$ p $的面孔一般位置。我们在$ j $ th级特征多项式的$ \ mathcal a $的$ j $ gp $的$ j $ faces数量中得出了一个公式。特别是，我们表明$ gp $的面部数不取决于线性地图$ g $，前提是满足一般位置假设。此外，我们为其在所有$ j $ faces的圆锥锥体的圆锥固有量和格拉曼角的总和提供了公式。我们将这些结果应用于$ a $ a $ a $ a $ a和$ b $的Persutohedra，这会以预计的Permutohedra的面部数量和Permutohedra的广义角度总和的封闭配方，从这种类型的stirling and and $ b $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ ANALOGUES方面。
Let $P\subset \mathbb R^n$ be a belt polytope, that is a polytope whose normal fan coincides with the fan of some hyperplane arrangement $\mathcal A$. Also, let $G:\mathbb R^n\to\mathbb R^d$ be a linear map of full rank whose kernel is in general position with respect to the faces of $P$. We derive a formula for the number of $j$-faces of the ``projected'' polytope $GP$ in terms of the $j$-th level characteristic polynomial of $\mathcal A$. In particular, we show that the face numbers of $GP$ do not depend on the linear map $G$ provided a general position assumption is satisfied. Furthermore, we derive formulas for the sum of the conic intrinsic volumes and Grassmann angles of the tangent cones of $P$ at all of its $j$-faces. We apply these results to permutohedra of types $A$ and $B$, which yields closed formulas for the face numbers of projected permutohedra and the generalized angle sums of permutohedra in terms of Stirling numbers of both kinds and their $B$-analogues.
