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David Roberts
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This is certainly one of the most vexing questions of all time! There really is no simple answer, though I agree that Allen has presented some reasonable ones.

In their paper Schubert calculus and representations of general linear group Mukhin-Tarasov-VarchenkoSchubert calculus and representations of general linear group (JAMS 22 Number 4 (2009) 909–940, doi:10.1090/S0894-0347-09-00640-7) Mukhin–Tarasov–Varchenko give another answer. They construct any algebra $ \mathcal{B} $ called the Bethe algebra, which acts on a tensor product multiplicity space $ Hom(V_\lambda, V_{\lambda_1} \otimes \cdots \otimes V_{\lambda_m}) $, depending on parameters $ b_1, \dots, b_n$. They prove that the image $ A $ of the Bethe algebra acting on this vector space has the same dimension of this vector space (for generic $ b_i $ you get all diagonal matrices with respect to some basis). Then they prove this algebra $ A $ is isomorphic to the functions on an scheme-theorectic intersection of $n+1 $ Schubert varieties corresponding to the $ \lambda_i$, with respect to flags given by the $ b_1, \dots, b_n $.

This is certainly one of the most vexing questions of all time! There really is no simple answer, though I agree that Allen has presented some reasonable ones.

In their paper Schubert calculus and representations of general linear group Mukhin-Tarasov-Varchenko give another answer. They construct any algebra $ \mathcal{B} $ called the Bethe algebra, which acts on a tensor product multiplicity space $ Hom(V_\lambda, V_{\lambda_1} \otimes \cdots \otimes V_{\lambda_m}) $, depending on parameters $ b_1, \dots, b_n$. They prove that the image $ A $ of the Bethe algebra acting on this vector space has the same dimension of this vector space (for generic $ b_i $ you get all diagonal matrices with respect to some basis). Then they prove this algebra $ A $ is isomorphic to the functions on an scheme-theorectic intersection of $n+1 $ Schubert varieties corresponding to the $ \lambda_i$, with respect to flags given by the $ b_1, \dots, b_n $.

This is certainly one of the most vexing questions of all time! There really is no simple answer, though I agree that Allen has presented some reasonable ones.

In their paper Schubert calculus and representations of general linear group (JAMS 22 Number 4 (2009) 909–940, doi:10.1090/S0894-0347-09-00640-7) Mukhin–Tarasov–Varchenko give another answer. They construct any algebra $ \mathcal{B} $ called the Bethe algebra, which acts on a tensor product multiplicity space $ Hom(V_\lambda, V_{\lambda_1} \otimes \cdots \otimes V_{\lambda_m}) $, depending on parameters $ b_1, \dots, b_n$. They prove that the image $ A $ of the Bethe algebra acting on this vector space has the same dimension of this vector space (for generic $ b_i $ you get all diagonal matrices with respect to some basis). Then they prove this algebra $ A $ is isomorphic to the functions on an scheme-theorectic intersection of $n+1 $ Schubert varieties corresponding to the $ \lambda_i$, with respect to flags given by the $ b_1, \dots, b_n $.

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Joel Kamnitzer
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This is certainly one of the most vexing questions of all time! There really is no simple answer, though I agree that Allen has presented some reasonable ones.

In their paper Schubert calculus and representations of general linear group Mukhin-Tarasov-Varchenko give another answer. They construct any algebra $ \mathcal{B} $ called the Bethe algebra, which acts on a tensor product multiplicity space $ Hom(V_\lambda, V_{\lambda_1} \otimes \cdots \otimes V_{\lambda_m}) $, depending on parameters $ b_1, \dots, b_n$. They prove that the image $ A $ of the Bethe algebra acting on this vector space has the same dimension of this vector space (for generic $ b_i $ you get all diagonal matrices with respect to some basis). Then they prove this algebra $ A $ is isomorphic to the functions on an scheme-theorectic intersection of $n+1 $ Schubert varieties corresponding to the $ \lambda_i$, with respect to flags given by the $ b_1, \dots, b_n $.