Two-step flag manifolds and the Horn conjecture
نویسنده
چکیده
We give a simplification of Belkale’s geometric proof of the Horn conjecture. Our approach uses the geometry of two-step flag manifolds to explain the occurrence of the Horn inequalities in a very straightforward way. The arguments for both necessity and sufficiency of the Horn inequalities are fairly conceptual when viewed in this framework. We provide examples to illustrate the method of proof.
منابع مشابه
The Puzzle Conjecture for the Cohomology of Two-step Flag Manifolds
We prove a conjecture of Knutson asserting that the Schubert structure constants of the cohomology ring of a two-step flag variety are equal to the number of puzzles with specified border labels that can be created using a list of eight puzzle pieces. As a consequence, we obtain a puzzle formula for the Gromov-Witten invariants defining the small quantum cohomology ring of a Grassmann variety o...
متن کاملEquivariant Mirrors and the Virasoro Conjecture for Flag Manifolds
We found an explicit description of allGL(n,R)-Whittaker functions as oscillatory integrals and thus constructed equivariant mirrors of flag manifolds. As a consequence we proved the Virasoro conjecture for flag manifolds.
متن کاملExperimentation and Conjectures in the Real Schubert Calculus for Flag Manifolds
The Shapiro conjecture in the real Schubert calculus, while likely true for Grassmannians, fails to hold for flag manifolds, but in a very interesting way. We give a refinement of the Shapiro conjecture for the flag manifold and present massive computational experimentation in support of this refined conjecture. We also prove the conjecture in some special cases using discriminants and establis...
متن کاملThe Abelian/nonabelian Correspondence and Frobenius Manifolds
We propose an approach via Frobenius manifolds to the study (began in [BCK2]) of the relation between rational Gromov-Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov-Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonab...
متن کاملThe Quantum Orbit Method for Generalized Flag Manifolds
Generalized flag manifolds endowed with the Bruhat-Poisson bracket are compact Poisson homogeneous spaces, whose decompositions in symplectic leaves coincide with their stratifications in Schubert cells. In this note it is proved that the irreducible ∗-representations of the corresponding quantized flag manifolds are also parametrized by their Schubert cells. An important step is the determinat...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008