Schubert calculus and representations of the general linear group

Mukhin, E.; Tarasov, V.; Varchenko, A.

doi:10.1090/S0894-0347-09-00640-7

Schubert calculus and representations of the general linear group
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by E. Mukhin, V. Tarasov and A. Varchenko

J. Amer. Math. Soc. 22 (2009), 909-940

DOI: https://doi.org/10.1090/S0894-0347-09-00640-7

Published electronically: April 30, 2009

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Abstract:

We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation $\mathfrak {gl}_N[t]$-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.

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