李群表示论和Schubert条件

本文将Grassmann流形上的Schubert子簇所满足的经典的Schubert条件推广到一般的复半单李群G的广义旗流形.利用复半单李群的表示理论,我们首先在李群的权空间上引入自然的Ehresman偏序.这一偏序可以导出李群的最高权表示的权系、Weyl群及其陪集空间上的Ehresman偏序.然后我们对一般的复表示定义了相应的射影空间,Grassmann流形和旗流形.这使得能够像经典的情形一样来分析广义旗流形的Schubert子簇满足的Schubert条件.在讨论中,我们还给出了李群G的Weyl群及其陪集空间中的Bruhat-Chevalley偏序的简单判别条件.我们的结果应用到例外群,给出了Fulton提出的关于例外群的Schubert分析的问题的部分回答.

Representation Theory and Schubert Condition

in this paper, we extend the classical Schubert conditions satisfied by Schubert subvarieties of Grassmannians to generalized flag manifolds of a complex semisimple Lie groups G. In the context of the representation theory of complex semisimple Lie group, we introduce Ehresman partial order on the weight system of Lie group G at first. This induces Ehresman partial orders on the weight systems of complex representations, Weyl group and it's coset spaces of G. Then we define projective spaces, Grasssmannians and flag manifolds correspondding to arbitrary complex representation of G, so that we can analyze the Schubert conditions satisfied by Schubert subvbarieties of generalized flag manifolds of G as in classical case. In our discussion, we also get a simple criteria for the Bruhat-Chevalley partial order in Weyl group and it's coset spaces. Apply our results to exceptional Lie groups, we obtain a partial answer to a problem on Schubert calculus of exceptional groups raised by Fulton.