Schubert calculus and representations of general linear group

2007/11/26 by E. Mukhin, Mukhin, E., V. Tarasov +3 · 5 citations
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.0711.4079

openalex publication_date 2007/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Jeu de taquin and a monodromy problem for Wronskians of polynomials
• Reality and transversality for Schubert calculus in OG(n,2n+1)
• On separation of variables and completeness of the Bethe ansatz for quantum gl_N Gaudin model
• Maximal transitivity of the cactus group on standard Young tableaux
• Bethe algebras for unitarizable modules over classical Lie (super)algebras and a duality

Related

• Intersections of Schubert varieties and highest weight vectors in tensor products of sl_N+1-representations
• Affine Hecke algebras and the Schubert calculus
• Representations from matrix varieties, and filtered RSK
• Schubert varieties and the fusion products
• The monodromy of real Bethe vectors for the Gaudin model