Total Positivity in Partial Flag Manifolds
نویسنده
چکیده
The projective space of Rn 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. Let G be a simply connected semisimple algebraic group over C with a fixed split R-structure. We will often identify a real algebraic variety with its set of R-rational points. This applies, in particular, to G and to the flag manifold B of G. In [L2] we have defined (in terms of an “épinglage” of G) the open subsemigroup G>0 of totally positive elements of G and a polyhedral open subset B>0 of B which in some sense plays the same role for G>0 as B for G. More generally, for any partial flag manifold PJ of G one can define the totally positive part PJ >0. (See [L4] or 1.5.) For J = ∅ we have PJ = B,PJ >0 = B>0. In this paper we show that PJ >0 is a connected component of an explicitly defined open real algebraic submanifold of PJ . We also show that, in the simply laced case, PJ >0 can be defined by algebraic inequalities involving canonical bases (see [L1]). These results confirm conjectures made in [L4]. In the special case where J = ∅, they reduce to known results from [L2].
منابع مشابه
Schubert Calculus and Puzzles
1. Interval positroid varieties 1 1.1. Schubert varieties 1 1.2. Schubert calculus 2 1.3. First positivity result 3 1.4. Interval rank varieties 5 2. Vakil’s Littlewood-Richardson rule 7 2.1. Combinatorial shifting 7 2.2. Geometric shifting 7 2.3. Vakil’s degeneration order 9 2.4. Partial puzzles 10 3. Equivariant and Kextensions 11 3.1. K-homology 11 3.2. K-cohomology 12 3.3. Equivariant K-the...
متن کاملRicci flow and manifolds with positive curvature
This is an expository article based on the author’s lecture delivered at the conference Lie Theory and Its Applications in March 2011, UCSD. We discuss various notions of positivity and their relations with the study of the Ricci flow, including a proof of the assertion, due to Wolfson and the author, that the Ricci flow preserves the positivity of the complex sectional curvature. We discuss th...
متن کاملQuantum Cohomology of Partial Flag Manifolds
We give elementary geometric proofs of the structure theorems for the (small) quantum cohomology of partial flag varieties SL(n)/P , including the quantum Pieri and quantum Giambelli formulas and the presentation.
متن کاملShadows of Characteristic Cycles, Verma Modules, and Positivity of Chern-schwartz-macpherson Classes of Schubert Cells
Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or noncompact varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex manifold. The theory of CSM classes has been extended to the equivariant setting by Ohmoto. We prove that for an arbitrary complex algebraic manifold X, the homogenized, torus equivariant CSM class of a constructibl...
متن کامل