The Full Kostant–Toda Hierarchy on the Positive Flag Variety
The Full Kostant–Toda Hierarchy on the Positive Flag Variety
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正旗品种上的完整 Kostant-Toda 层次结构
DOI：
10.1007/s00220-014-2203-x
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发表时间：
2013
影响因子：
2.4
通讯作者：
L. Williams
中科院分区：
文献类型：
--
作者：
Y. Kodama;L. Williams
当初始数据由$${SL_n（\mathbb{R}）}$$SLn（R）的完全非负（tnn）标志簇上的任意点给出时，我们研究了完整Kostant-Toda（f-KT）族解的几何和组合方面. tnn旗簇上的f-KT流是完备的，并且证明了它们的渐近性完全由Rietsch给出的tnn旗簇的胞腔分解决定（Total positivity and真实的flag varieties.博士论文，马萨诸塞州理工学院，剑桥，1998年）。我们的结果代表的f-KT层次（甚至f-KT格）的渐近性的第一个结果，此外，我们的结果不限于通用流，但涵盖非通用流以及。我们通过矩映射定义了权空间上的f-KT流，并证明了每个f-KT流的闭包构成一个有趣的凸多面体，我们称之为Bruhat区间多面体.特别地，一般流的Bruhat区间多面体是对称群$${\mathfrak{S}_n}$$Sn的置换面体。我们还证明了类似的结果，充分对称户田族，通过映射我们的f-KT解决方案的充分对称户田族。在附录中，我们表明，Bruhat区间多面体是广义置换面，在Postnikov（国际数学研究。IMRN（6）：1026-1106，2009）。
We study some geometric and combinatorial aspects of the solution to the full Kostant–Toda (f-KT) hierarchy, when the initial data is given by an arbitrary point on the totally non-negative (tnn) flag variety of $${SL_n(\mathbb{R})}$$SLn(R) . The f-KT flows on the tnn flag variety are complete, and we show that their asymptotics are completely determined by the cell decomposition of the tnn flag variety given by Rietsch (Total positivity and real flag varieties. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, 1998). Our results represent the first results on the asymptotics of the f-KT hierarchy (and even the f-KT lattice); moreover, our results are not confined to the generic flow, but cover non-generic flows as well. We define the f-KT flow on the weight space via the moment map, and show that the closure of each f-KT flow forms an interesting convex polytope which we call a Bruhat interval polytope. In particular, the Bruhat interval polytope for the generic flow is the permutohedron of the symmetric group $${\mathfrak{S}_n}$$Sn . We also prove analogous results for the full symmetric Toda hierarchy, by mapping our f-KT solutions to those of the full symmetric Toda hierarchy. In the appendix we show that Bruhat interval polytopes are generalized permutohedra, in the sense of Postnikov (Int. Math. Res. Not. IMRN (6):1026–1106, 2009).
影响因子：
1.3
作者：
Federico Ardila;F. Rincón;L. Williams
通讯作者：
Federico Ardila;F. Rincón;L. Williams