Secants of minuscule and cominuscule minimal orbits

Laurent Manivel; Mateusz Michalek

doi:10.1016/j.laa.2015.04.027

Secants of minuscule and cominuscule minimal orbits

Laurent Manivel^*, Mateusz Michalek

^*Corresponding author for this work

Not published at Aalto University

Research output: Contribution to journal › Article › Scientific › peer-review

Abstract

We study the geometry of the secant and tangential variety of a cominuscule and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods inspired by statistics we provide an explicit local isomorphism with a product of an affine space with a variety which is the Zariski closure of the image of a map defined by generalized determinants. In particular, equations of the secant or tangential variety correspond to relations among generalized determinants. We also provide a representation theoretic decomposition of cubics in the ideal of the secant variety of any Grassmannian. (C) 2015 Elsevier Inc. All rights reserved.

Original language	English
Pages (from-to)	288-312
Number of pages	25
Journal	Linear Algebra and Its Applications
Volume	481
DOIs	https://doi.org/10.1016/j.laa.2015.04.027
Publication status	Published - 15 Sept 2015
MoE publication type	A1 Journal article-refereed

Funding

The second author is supported by the Narodowe Centrum Nauki grant UMO-2011/01/N/ST1/05424 "Representation theory and secants of homogeneous varieties".

Keywords

Grassmannian
Minuscule and cominuscule representation
Secant variety
Generalized determinant
Cumulants
TANGENTIAL VARIETIES

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10.1016/j.laa.2015.04.027

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title = "Secants of minuscule and cominuscule minimal orbits",

abstract = "We study the geometry of the secant and tangential variety of a cominuscule and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods inspired by statistics we provide an explicit local isomorphism with a product of an affine space with a variety which is the Zariski closure of the image of a map defined by generalized determinants. In particular, equations of the secant or tangential variety correspond to relations among generalized determinants. We also provide a representation theoretic decomposition of cubics in the ideal of the secant variety of any Grassmannian. (C) 2015 Elsevier Inc. All rights reserved.",

keywords = "Grassmannian, Minuscule and cominuscule representation, Secant variety, Generalized determinant, Cumulants, TANGENTIAL VARIETIES",

author = "Laurent Manivel and Mateusz Michalek",

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day = "15",

doi = "10.1016/j.laa.2015.04.027",

language = "English",

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T1 - Secants of minuscule and cominuscule minimal orbits

AU - Manivel, Laurent

AU - Michalek, Mateusz

PY - 2015/9/15

Y1 - 2015/9/15

N2 - We study the geometry of the secant and tangential variety of a cominuscule and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods inspired by statistics we provide an explicit local isomorphism with a product of an affine space with a variety which is the Zariski closure of the image of a map defined by generalized determinants. In particular, equations of the secant or tangential variety correspond to relations among generalized determinants. We also provide a representation theoretic decomposition of cubics in the ideal of the secant variety of any Grassmannian. (C) 2015 Elsevier Inc. All rights reserved.

AB - We study the geometry of the secant and tangential variety of a cominuscule and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods inspired by statistics we provide an explicit local isomorphism with a product of an affine space with a variety which is the Zariski closure of the image of a map defined by generalized determinants. In particular, equations of the secant or tangential variety correspond to relations among generalized determinants. We also provide a representation theoretic decomposition of cubics in the ideal of the secant variety of any Grassmannian. (C) 2015 Elsevier Inc. All rights reserved.

KW - Grassmannian

KW - Minuscule and cominuscule representation

KW - Secant variety

KW - Generalized determinant

KW - Cumulants

KW - TANGENTIAL VARIETIES

U2 - 10.1016/j.laa.2015.04.027

DO - 10.1016/j.laa.2015.04.027

M3 - Article

SN - 1873-1856

VL - 481

SP - 288

EP - 312

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Secants of minuscule and cominuscule minimal orbits

Abstract

Funding

Keywords

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