Schubert calculus and representations of the general linear group

Abstract

We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation g l N [ t ] \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.

A congruence modulo four in real Schubert calculus

“…. , t n ) is a real point in (P 1 ) n = , then each t i is real and the Mukhin-Tarasov-Varchenko Theorem [19,20] states that the intersection (4.1) consists of d(λ) real points, so that the desired congruence to d(λ) modulo four is not interesting.…”

“…Significant evidence for this conjecture, both theoretical and computational, was found in [28]. Eremenko and Gabrielov [5] proved the conjecture when min{m, p} = 2, and it was proved for all m, p by Mukhin, Tarasov, and Varchenko [19,20]. In their study of this conjecture, Eremenko and Gabrielov [3] calculated the topological degree of the real Wronski map, obtaining a nonzero lower bound for the number of real subspaces in the fiber above a general real polynomial of degree mp, when m+p is odd.…”

Когомологии Многообразия Флагов Как Алгебра Бете

“…sing λ в работе [9]. Последняя была использована в [9] для получения геометрического описания действия алгеб-ры B K=0 в пространстве (V + ) sing λ .…”

“…Последняя была использована в [9] для получения геометрического описания действия алгеб-ры B K=0 в пространстве (V + ) sing λ . Используя формулу (3.4), можно получить аналогичное геометрическое описание действия алгебры B K=0 в пространстве (…”

“…[9]. Эта интерпретация устанавливает соответствие между теорией представлений и исчислением Шуберта, полезное для обеих сторон.…”