Wronskians、循环群动作和Ribbon tableaux
Wronskians, cyclic group actions, and ribbon tableaux
摘要：
Wronski映射是从Grassmannian Gr（d，n）到射影空间（多项式向量空间的射影化）的有限PGL_2（C）-等变态射。我们考虑以下问题。若C_r < PGL_2（C）是r阶循环子群，则C_r-不动点在基上的Wronski映射的纤维中如何存在？在本文中，我们计算的一般答案的r-带状tableaux。当r=2时，该计算给出了具有纯虚根的实多项式上的Wronski映射的纤维中的实点数。更一般地，我们可以计算舒伯特簇的某些交叉中的实点的数量。当r整除d（n-d）时，我们的主要结果是：纤维中C_r-不动点的一般个数是标准r-带状tableaux矩形的个数（n-d）^d。通过不同的方法计算，我们表明，在这种情况下的答案也是由形状为（n-d）^d的标准Young tableaux的数量给出的，这些标准Young tableaux在Jeu de taquin提升的N/r次迭代下不变。这两个结果共同给出了Rhoades推广循环筛选定理在矩形tableaux上的一个新的证明。证明了二面体群作用的类似结果。
英文摘要：
The Wronski map is a finite, PGL_2(C)-equivariant morphism from the Grassmannian Gr(d,n) to a projective space (the projectivization of a vector space of polynomials). We consider the following problem. If C_r < PGL_2(C) is a cyclic subgroup of order r, how may C_r-fixed points are in the in a fibre of the Wronski map over a C_r-fixed point in the base? In this paper, we compute a general answer in terms of r-ribbon tableaux. When r=2, this computation gives the number of real points in the fibre of the Wronski map over a real polynomial with purely imaginary roots. More generally, we can compute the number of real points in certain intersections of Schubert varieties. When r divides d(n-d) our main result says that the generic number of C_r-fixed points in the fibre is the number of standard r-ribbon tableaux rectangular shape (n-d)^d. Computing by a different method, we show that the answer in this case is also given by the number of of standard Young tableaux of shape (n-d)^d that are invariant under N/r iterations of jeu de taquin promotion. Together, these two results give a new proof of Rhoades' cyclic sieving theorem for promotion on rectangular tableaux. We prove analogous results for dihedral group actions.
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