### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 046, 15 pages      arXiv:1710.06534      https://doi.org/10.3842/SIGMA.2018.046

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

Kang Lu
Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA

Received November 27, 2017, in final form May 07, 2018; Published online May 14, 2018

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.

Key words: real Schubert calculus; self-dual spaces; Bethe ansatz; Gaudin model.

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