Tensor invariants, saturation problems, and Dynkin automorphisms

Jiuzu Hong; Linhui Shen

doi:10.1016/j.aim.2015.08.015

Tensor invariants, saturation problems, and Dynkin automorphisms

Jiuzu Hong^*, Linhui Shen

^*Corresponding author for this work

Mathematics

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

7 Scopus citations

Abstract

Let G be a connected almost simple algebraic group with a Dynkin automorphism σ. Let G_σ be the connected almost simple algebraic group associated with G and σ. We prove that the dimension of the tensor invariant space of G_σ is equal to the trace of σ on the corresponding tensor invariant space of G. We prove that if G has the saturation property then so does G_σ. As a consequence, we show that the spin group Spin(2n+1) has saturation factor 2, which strengthens the results of Belkale-Kumar [1] and Sam [28] in the case of type B_n.

Original language	English (US)
Pages (from-to)	629-657
Number of pages	29
Journal	Advances in Mathematics
Volume	285
DOIs	https://doi.org/10.1016/j.aim.2015.08.015
State	Published - Nov 5 2015

Keywords

Affine Grassmannian
Geometric Satake correspondence
Satake basis
Saturation problems
Tensor multiplicity
Tropical points
Twining formula

ASJC Scopus subject areas

Mathematics(all)

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10.1016/j.aim.2015.08.015

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title = "Tensor invariants, saturation problems, and Dynkin automorphisms",

abstract = "Let G be a connected almost simple algebraic group with a Dynkin automorphism σ. Let Gσ be the connected almost simple algebraic group associated with G and σ. We prove that the dimension of the tensor invariant space of Gσ is equal to the trace of σ on the corresponding tensor invariant space of G. We prove that if G has the saturation property then so does Gσ. As a consequence, we show that the spin group Spin(2n+1) has saturation factor 2, which strengthens the results of Belkale-Kumar [1] and Sam [28] in the case of type Bn.",

keywords = "Affine Grassmannian, Geometric Satake correspondence, Satake basis, Saturation problems, Tensor multiplicity, Tropical points, Twining formula",

author = "Jiuzu Hong and Linhui Shen",

note = "Publisher Copyright: {\textcopyright} 2015 Elsevier Inc.",

year = "2015",

month = nov,

day = "5",

doi = "10.1016/j.aim.2015.08.015",

language = "English (US)",

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TY - JOUR

T1 - Tensor invariants, saturation problems, and Dynkin automorphisms

AU - Hong, Jiuzu

AU - Shen, Linhui

PY - 2015/11/5

Y1 - 2015/11/5

N2 - Let G be a connected almost simple algebraic group with a Dynkin automorphism σ. Let Gσ be the connected almost simple algebraic group associated with G and σ. We prove that the dimension of the tensor invariant space of Gσ is equal to the trace of σ on the corresponding tensor invariant space of G. We prove that if G has the saturation property then so does Gσ. As a consequence, we show that the spin group Spin(2n+1) has saturation factor 2, which strengthens the results of Belkale-Kumar [1] and Sam [28] in the case of type Bn.

AB - Let G be a connected almost simple algebraic group with a Dynkin automorphism σ. Let Gσ be the connected almost simple algebraic group associated with G and σ. We prove that the dimension of the tensor invariant space of Gσ is equal to the trace of σ on the corresponding tensor invariant space of G. We prove that if G has the saturation property then so does Gσ. As a consequence, we show that the spin group Spin(2n+1) has saturation factor 2, which strengthens the results of Belkale-Kumar [1] and Sam [28] in the case of type Bn.

KW - Affine Grassmannian

KW - Geometric Satake correspondence

KW - Satake basis

KW - Saturation problems

KW - Tensor multiplicity

KW - Tropical points

KW - Twining formula

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DO - 10.1016/j.aim.2015.08.015

M3 - Article

AN - SCOPUS:84940534547

SN - 0001-8708

VL - 285

SP - 629

EP - 657

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

Tensor invariants, saturation problems, and Dynkin automorphisms

Abstract

Keywords

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