The point variety of quantum polynomial rings

An immanant formulation of the dual canonical basis of the quantum polynomial ring

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2703 ◽

2009 ◽

Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽

Cited By ~ 1

Author(s):

Mark Skandera ◽

Justin Lambright

Keyword(s):

Polynomial Ring ◽

Canonical Basis ◽

Polynomial Rings ◽

Dual Canonical Basis ◽

International Audience ◽

Quantum Polynomial ◽

Lusztig Polynomials

International audience We show that dual canonical basis elements of the quantum polynomial ring in $n^2$ variables can be expressed as specializations of dual canonical basis elements of $0$-weight spaces of other quantum polynomial rings. Our results rely upon the natural appearance in the quantum polynomial ring of Kazhdan-Lusztig polynomials, $R$-polynomials, and certain single and double parabolic generalizations of these. Nous démontrons que des éléments de la base canonique duale de l'anneau quantique des polynômes en $n^2$ variables peuvent s'exprimer en termes des spécialisations d'éléments de la base canonique duale des espaces de poids $0$ d'autres anneaux quantiques. Nos résultats dépendent fortement de l'apparition naturelle des polynômes de Kazhdan-Lusztig, des $R$-polynômes, et de certaines généralisations simplement et doublement paraboliques de ces polynômes dans l'anneau quantique.

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Automorphisms for SkewPBWExtensions and Skew Quantum Polynomial Rings

Communications in Algebra ◽

10.1080/00927872.2013.879163 ◽

2015 ◽

Vol 43 (5) ◽

pp. 1877-1897 ◽

Cited By ~ 3

Author(s):

César Fernando Venegas Ramírez

Keyword(s):

Polynomial Rings ◽

Quantum Polynomial

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Quantum Drinfeld Hecke Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-2013-012-2 ◽

2014 ◽

Vol 66 (4) ◽

pp. 874-901 ◽

Cited By ~ 8

Author(s):

Viktor Levandovskyy ◽

Anne V. Shepler

Keyword(s):

Hecke Algebras ◽

Sufficient Conditions ◽

Polynomial Rings ◽

Skew Polynomial Rings ◽

Symplectic Reflection Algebras ◽

Quantum Polynomial ◽

Reflection Algebras ◽

Coordinate Rings ◽

Necessary And Sufficient ◽

Skew Polynomial

AbstractWe consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincaré–Birkhoff–Witt property using the theory of noncommutative Gröbner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the Poincaré–Birkhoff–Witt conditions.

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