On commuting varieties of parabolic subalgebras

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On commuting varieties of parabolic subalgebras. / Goodwin, Simon; Goddard, Russell.

In: Journal of Pure and Applied Algebra, 19.04.2017.

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@article{e48d4c37b54d4783904fd6779a86a14c,
title = "On commuting varieties of parabolic subalgebras",
abstract = "Let G be a connected reductive algebraic group over an algebraically closed field k, and assume that the characteristic of k is zero or a pretty good prime for G. Let P be a parabolic subgroup of G and let p be the Lie algebra of P. We consider the commuting variety C(p)={(X,Y)∈p×p∣[X,Y]=0}. Our main theorem gives a necessary and sufficient condition for irreducibility of C(p) in terms of the modality of the adjoint action of P on the nilpotent variety of p. As a consequence, for the case P=B a Borel subgroup of G, we give a classification of when C(b) is irreducible; this builds on a partial classification given by Keeton. Further, in cases where C(p) is irreducible, we consider whether C(p) is a normal variety. In particular, this leads to a classification of when C(b) is normal.",
author = "Simon Goodwin and Russell Goddard",
year = "2017",
month = apr,
day = "19",
doi = "10.1016/j.jpaa.2017.04.015",
language = "English",
journal = "Journal of Pure and Applied Algebra",
issn = "0022-4049",
publisher = "Elsevier",

}

RIS

TY - JOUR

T1 - On commuting varieties of parabolic subalgebras

AU - Goodwin, Simon

AU - Goddard, Russell

PY - 2017/4/19

Y1 - 2017/4/19

N2 - Let G be a connected reductive algebraic group over an algebraically closed field k, and assume that the characteristic of k is zero or a pretty good prime for G. Let P be a parabolic subgroup of G and let p be the Lie algebra of P. We consider the commuting variety C(p)={(X,Y)∈p×p∣[X,Y]=0}. Our main theorem gives a necessary and sufficient condition for irreducibility of C(p) in terms of the modality of the adjoint action of P on the nilpotent variety of p. As a consequence, for the case P=B a Borel subgroup of G, we give a classification of when C(b) is irreducible; this builds on a partial classification given by Keeton. Further, in cases where C(p) is irreducible, we consider whether C(p) is a normal variety. In particular, this leads to a classification of when C(b) is normal.

AB - Let G be a connected reductive algebraic group over an algebraically closed field k, and assume that the characteristic of k is zero or a pretty good prime for G. Let P be a parabolic subgroup of G and let p be the Lie algebra of P. We consider the commuting variety C(p)={(X,Y)∈p×p∣[X,Y]=0}. Our main theorem gives a necessary and sufficient condition for irreducibility of C(p) in terms of the modality of the adjoint action of P on the nilpotent variety of p. As a consequence, for the case P=B a Borel subgroup of G, we give a classification of when C(b) is irreducible; this builds on a partial classification given by Keeton. Further, in cases where C(p) is irreducible, we consider whether C(p) is a normal variety. In particular, this leads to a classification of when C(b) is normal.

U2 - 10.1016/j.jpaa.2017.04.015

DO - 10.1016/j.jpaa.2017.04.015

M3 - Article

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

ER -