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A quantum analogue of the dihedral action on Grassmannians

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A quantum analogue of the dihedral action on Grassmannians. / Allman, Justin M.; Grabowski, Jan.
In: Journal of Algebra, Vol. 359, No. 1, 06.2012, p. 49-68.

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Allman JM, Grabowski J. A quantum analogue of the dihedral action on Grassmannians. Journal of Algebra. 2012 Jun;359(1):49-68. Epub 2012 Mar 28. doi: 10.1016/j.jalgebra.2012.03.016

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Allman, Justin M. ; Grabowski, Jan. / A quantum analogue of the dihedral action on Grassmannians. In: Journal of Algebra. 2012 ; Vol. 359, No. 1. pp. 49-68.

Bibtex

@article{8060e97e5bbb43ad891026eaa8da82de,
title = "A quantum analogue of the dihedral action on Grassmannians",
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.",
keywords = "Quantum Grassmannian, Twisting , Dihedral group",
author = "Allman, {Justin M.} and Jan Grabowski",
year = "2012",
month = jun,
doi = "10.1016/j.jalgebra.2012.03.016",
language = "English",
volume = "359",
pages = "49--68",
journal = "Journal of Algebra",
issn = "0021-8693",
publisher = "ELSEVIER ACADEMIC PRESS INC",
number = "1",

}

RIS

TY - JOUR

T1 - A quantum analogue of the dihedral action on Grassmannians

AU - Allman, Justin M.

AU - Grabowski, Jan

PY - 2012/6

Y1 - 2012/6

N2 - 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.

AB - 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.

KW - Quantum Grassmannian

KW - Twisting

KW - Dihedral group

U2 - 10.1016/j.jalgebra.2012.03.016

DO - 10.1016/j.jalgebra.2012.03.016

M3 - Journal article

VL - 359

SP - 49

EP - 68

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -