2008
DOI: 10.1017/s0017089507003928
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Quantum Analogues of Schubert Varieties in the Grassmannian

Abstract: Abstract. We study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains that are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras a… Show more

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Cited by 10 publications
(33 citation statements)
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“…It is the subalgebra of O q (M v,u (k)) generated by the u × u quantum minors of O q (M v,u (k)). Notice that we adopt here a convention exchanging rows and columns with respect to the convention of [LR1] and [LR2]. However, embedding all the relevant algebras in O q (M v (k)) and using the transpose automorphism introduced above shows that the two different conventions lead to isomorphic algebras.…”
Section: Quantum Analogues Of Richardson Varietiesmentioning
confidence: 99%
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“…It is the subalgebra of O q (M v,u (k)) generated by the u × u quantum minors of O q (M v,u (k)). Notice that we adopt here a convention exchanging rows and columns with respect to the convention of [LR1] and [LR2]. However, embedding all the relevant algebras in O q (M v (k)) and using the transpose automorphism introduced above shows that the two different conventions lead to isomorphic algebras.…”
Section: Quantum Analogues Of Richardson Varietiesmentioning
confidence: 99%
“…Namely, the fact that quantum analogues of Richardson varieties degenerate to quantum analogues of toric varieties. (A much more elementary proof of the fact that quantum analogues of Schubert varities are integral domains is given in [LR2]. )…”
Section: Introductionmentioning
confidence: 99%
“…Most proofs will be omitted since these results already appear in [12,15,16]. Appropriate references will be given in the text.…”
Section: Basic Definitionsmentioning
confidence: 99%
“…The quantum Schubert variety S(γ) associated to γ is S(γ) := O q (G m,n (k))/ Π γ m,n . (Note that S(γ) was denoted by O q (G m,n (k)) γ in [16].) This definition is inspired by the classical description of the coordinate rings of Schubert varieties in the grassmannian.…”
Section: Basic Definitionsmentioning
confidence: 99%
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