Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus

Abstract

The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin model associated to the Lie algebras of types B and C. In this paper, we give lower bounds for the numbers of real self-dual spaces in intersections of Schubert varieties related to osculating flags in the Grassmannian. The higher Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of this form.

In Sections 4 and 5, we recall the definition of self-dual spaces and the relations between self-dual spaces and Gaudin model in types B and C from arXiv:1705.02048. We use similar strategy (but in a different context) and exposition style as in the recent work arXiv:1404.7194 of Mukhin and Tarasov

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• Schubert calculus and representations of general linear group
• On the Gaudin model associated to Lie algebras of classical types
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• Eigenvalues of Bethe vectors in the Gaudin model
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Cited by in corpus (2)

• Bethe ansatz equations for orthosymplectic Lie superalgebras and self-dual superspaces
• Perfect Integrability and Gaudin Models