Total positivity in partial flag manifolds. (English) Zbl 0895.14014
Summary: The projective space of \(\mathbb{R}^n\) has a natural open subset: the set of lines spanned by vectors with all coordinates \(>0\). Such a subset can be defined more generally for any partial flag manifold of a split semisimple real algebraic group. The main result of the paper is that this subset can be defined by algebraic equalities and inequalities.
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |
References:
| [1] | George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. · Zbl 0788.17010 |
| [2] | G. Lusztig, Total positivity in reductive groups, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 531 – 568. · Zbl 0845.20034 |
| [3] | G. Lusztig, Total positivity and canonical bases, Algebraic groups and Lie groups , Cambridge Univ. Press, 1997, pp. 281-295. · Zbl 0890.20034 |
| [4] | G. Lusztig, Introduction to total positivity, Positivity in Lie theory: open problems, De Gruyter (to appear). · Zbl 0929.20035 |
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