Schubert calculus and representations of general linear group

Abstract

We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation gl_N[t]-modules and the scheme-theoretic intersection of suitable Schubert varieties. Moreover, we prove that the multiplicity space as a module over the Bethe algebra is isomorphic to the coregular representation of the scheme-theoretic intersection. In particular, this result implies the simplicity of the spectrum of the Bethe algebra for real values of evaluation parameters and the transversality of the intersection of the corresponding Schubert varieties.

Latex, 32 pages

References in corpus (5)

• Quantization of the Gaudin System
• Current algebras, highest weight categories and quivers
• The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
• Multiple orthogonal polynomials and a counterexample to Gaudin Bethe Ansatz Conjecture
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• Minimal Kinematics: An All $k$ and $n$ Peek into ${\mathrm{Trop}}^{+}\mathrm{G}(k,n)$
• Three sides of the geometric Langlands correspondence for gl_N Gaudin model and Bethe vector averaging maps
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• Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus
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