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A quantum analogue of the dihedral action on Grassmannians

Allman, Justin M. and Grabowski, Jan (2012) A quantum analogue of the dihedral action on Grassmannians. Journal of Algebra, 359 (1). pp. 49-68. ISSN 0021-8693

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Abstract

In recent work, Launois and Lenagan have shown how to construct a cocycle twisting of the quantum Grassmannian and an isomorphism of the twisted and untwisted algebras that sends a given quantum minor to the minor whose index set is permuted according to the $n$-cycle $c=(1\,2\, \cdots \,n)$, up to a power of $q$. This twisting is needed because $c$ does not naturally induce an automorphism of the quantum Grassmannian, as it does classically and semi-classically. We extend this construction to give a quantum analogue of the action on the Grassmannian of the dihedral subgroup of $S_{n}$ generated by $c$ and $w_{0}$, the longest element, and this analogue takes the form of a groupoid. We show that there is an induced action of this subgroup on the torus-invariant prime ideals of the quantum Grassmannian and also show that this subgroup acts on the totally nonnegative and totally positive Grassmannians. Then we see that this dihedral subgroup action exists classically, semi-classically (by Poisson automorphisms and anti-automorphisms, a result of Yakimov) and in the quantum and nonnegative settings.

Item Type: Article
Journal or Publication Title: Journal of Algebra
Uncontrolled Keywords: Quantum Grassmannian ; Twisting ; Dihedral group
Subjects: Q Science > QA Mathematics
Departments: Faculty of Science and Technology > Mathematics and Statistics
ID Code: 53715
Deposited By: ep_importer_pure
Deposited On: 20 Apr 2012 13:22
Refereed?: Yes
Published?: Published
Last Modified: 09 Apr 2014 23:21
Identification Number:
URI: http://eprints.lancs.ac.uk/id/eprint/53715

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