Prime ideals in the quantum grassmannian

Launois, Stéphane; Lenagan, T. H.; Rigal, Laurent

doi:10.1007/s00029-008-0054-z

Cited by 24 publications

(50 citation statements)

References 19 publications

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“…• prime ideals in noncommutative deformations of G/P (though worked out only for the Grassmannian, in [LauLeRig08]), and a semiclassical version thereof in Poisson geometry [BroGooYa06,GooYa]; • the characteristic p notion of Frobenius splitting ([KnLamSp]).…”

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confidence: 99%

Positroid varieties: juggling and geometry

Knutson¹,

Lam²,

Speyer³

2013

Compositio Math.

168

302

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While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown-Goodearl-Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen-Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov's and Buch-Kresch-Tamvakis' approaches to quantum Schubert calculus.

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mentioning

confidence: 99%

Positroid varieties: juggling and geometry

Knutson¹,

Lam²,

Speyer³

2013

Compositio Math.

168

302

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“…It is well known that K q [Gr(k, n)] is a Noetherian domain with Gel'fand-Kirillov dimension k(n − k) + 1. (Further ring-theoretic properties of quantum Grassmannians are established in [6,18,21,22]. )…”

Section: Quantum Matrices and Quantum Grassmanniansmentioning

confidence: 99%

Graded quantum cluster algebras and an application to quantum Grassmannians

Grabowski

Launois

2014

Proceedings of the London Mathematical Society

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We introduce a framework for Z-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation. The resulting grading has the property that every (quantum) cluster variable is homogeneous.In the quantum setting, we use this grading framework to give a construction that behaves somewhat like twisting, in that it produces a new quantum cluster algebra with the same cluster combinatorics but with different quasi-commutation relations between the cluster variables.We apply these results to show that the quantum Grassmannians Kq[Gr(k, n)] admit quantum cluster algebra structures, as quantizations of the cluster algebra structures on the classical Grassmannian coordinate ring found by Scott. This is done by lifting the quantum cluster algebra structure on quantum matrices due to Geiß-Leclerc-Schröer and completes earlier work of the authors on the finite-type cases.

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“…It turns out that, as long as the deformation parameters are chosen in a sufficiently generic manner, G-Spec R is indeed finite for all quantized coordinate algebras R = O q (X) that have been analyzed in detail thus far, the acting group G typically being a suitably chosen algebraic torus. Notable examples include the (generic) quantized coordinate rings of all semisimple algebraic groups (Joseph [13], Hodges, Levasseur and Toro [11]), quantum matrices and quantum Grassmannians (Cauchon, Lenagan and others; e.g, [7], [8] and [17]). Finiteness of G-Spec R has also been observed in Leavitt path algebras R, again for the action of a suitable torus G [1].…”

Section: This Article Addresses the Following General Questionmentioning

confidence: 99%

Rational Group Actions on Affine Pi-Algebras

Lorenz

2013

Glasgow Math. J.

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ABSTRACT. Let R be an affine PI-algebra over an algebraically closed field and let G be an affine algebraic -group that acts rationally by algebra automorphisms on R. For R prime and G a torus, we show that R has only finitely many G-prime ideals if and only if the action of G on the center of R is multiplicity free. This extends a standard result on affine algebraic G-varieties. Under suitable hypotheses on R and G, we also prove a PI-version of a well-known result on spherical varieties and a version of Schelter's catenarity theorem for G-primes.

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Prime ideals in the quantum grassmannian

Cited by 24 publications

References 19 publications

Positroid varieties: juggling and geometry

Positroid varieties: juggling and geometry

Graded quantum cluster algebras and an application to quantum Grassmannians

Rational Group Actions on Affine Pi-Algebras

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