On The Free and G-Saturated Weight Monoids of Smooth Affine Spherical Varieties For G=SL(n)

123 Pages·2017·0.61 MB·English

CCiittyy UUnniivveerrssiittyy ooff NNeeww YYoorrkk ((CCUUNNYY))
CCUUNNYY AAccaaddeemmiicc WWoorrkkss
Dissertations, Theses, and Capstone Projects CUNY Graduate Center
9-2016
OOnn TThhee FFrreeee aanndd GG--SSaattuurraatteedd WWeeiigghhtt MMoonnooiiddss ooff SSmmooootthh AAffiffinnee
SSpphheerriiccaall VVaarriieettiieess FFoorr GG==SSLL((nn))
Won Geun Kim
The Graduate Center, City University of New York
How does access to this work benefit you? Let us know!
More information about this work at: https://academicworks.cuny.edu/gc_etds/1598
Discover additional works at: https://academicworks.cuny.edu
This work is made publicly available by the City University of New York (CUNY).
Contact: [email protected]
ON THE FREE AND G-SATURATED
WEIGHT MONOIDS OF SMOOTH AFFINE
SPHERICAL VARIETIES FOR G = SL(n)
by
Won Geun Kim
A dissertation submitted to the Graduate Faculty in Mathematics in partial
fulfillment of the requirements for the degree of Doctor of Philosophy, The
City University of New York.
2016
ii
(cid:13)c 2016
Won Geun Kim
All Rights Reserved
iii
This manuscript has been read and accepted for the Graduate Faculty in
Mathematics in satisfaction of the dissertation requirements for the degree
of Doctor of Philosophy.
Gautam Chinta
Date Chair of Examining Committee
Ara Basmajian
Date Executive Officer
Raymond Hoobler
Benjamin Steinberg
Bart Van Steirteghem : co-advisor
Supervisory Committee
THE CITY UNIVERSITY OF NEW YORK
iv
Abstract
ON THE FREE AND G-SATURATED WEIGHT MONOIDS
OF SMOOTH AFFINE SPHERICAL VARIETIES FOR
G = SL(n)
by
Won Geun Kim
Advisor: Gautam Chinta and Bart Van Steirteghem
Let X be an affine algebraic variety over C equipped with an action
of a connected reductive group G. The weight monoid Γ(X) of X is the
set of isomorphism classes of irreducible representations of G that occur in
the coordinate ring C[X] of X. Losev has shown that if X is a smooth
affine spherical variety, that is, if X is smooth and C[X] is multiplicity-
free as a representation of G, then Γ(X) determines X up to equivariant
automorphism.
PezziniandVanSteirteghemhaverecentlyobtainedacombinatorialchar-
acterization of the weight monoids of smooth affine spherical varieties, using
the combinatorial theory of spherical varieties and a smoothness criterion
due to R. Camus. The first part of this thesis gives an implementation in
Sage of a special case of this combinatorial characterization: given a free and
“G-saturated” monoid Γ of dominant weights for G = SL(n), the algorithm
decides whether there exists a smooth affine spherical G-variety X such that
Γ(X) = Γ.
In the second part of the thesis, we apply Pezzini and Van Steirteghem’s
characterizationtodeterminewhichsubsetsofthesetoffundamentalweights
of SL(n) generate a monoid that is the weight monoid of a smooth affine
spherical G-variety.
Acknowledgments
First and foremost, I would like to thank my advisors, Gautam Chinta and
Bart Van Steirteghem. Extremely gifted teachers, they have been a constant
source of inspiration. Their rigor, determination and kindness have kept this
project on track. My special thanks go to Ray Hoobler, a real mentor. If
I had not met him at the City College of New York it is very likely that I
wouldnothavestudiedmathematics. IamgratefultoBenjaminSteinbergfor
his generosity of serving as a committee member and for useful discussions.
Thanks to Guido Pezzini for suggesting the problem solved in Chapter 4 and
for other helpful suggestions. Thanks to Sage Days 65 and GAeL XXIV for
funding support. Last but not least, I would also like to thank my wife,
Nina, and family in the US and Korea for their words of advice and their
encouragement in my study and life.
v
Contents
Introduction 1
1 Preliminaries 4
1.1 The Classification of Spherical Varieties . . . . . . . . . . . . . 4
1.2 Combinatorial Invariants of Affine Spherical Varieties . . . . . 5
1.2.1 The Weight Monoid . . . . . . . . . . . . . . . . . . . 6
1.2.2 N-Spherical Roots . . . . . . . . . . . . . . . . . . . . 8
2 Algorithm and Implementation 10
2.1 Combinatorial Invariants . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 The Inputs . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Subalgorithm I . . . . . . . . . . . . . . . . . . . . . . 13
2.1.3 Subalgorithm II . . . . . . . . . . . . . . . . . . . . . . 15
2.1.4 Subalgorithm III . . . . . . . . . . . . . . . . . . . . . 27
2.1.5 Subalgorithm IV . . . . . . . . . . . . . . . . . . . . . 48
2.1.6 The Output . . . . . . . . . . . . . . . . . . . . . . . . 54
2.2 Implementation of Algorithm . . . . . . . . . . . . . . . . . . 54
3 Examples 57
3.1 The Weight Monoid of SL(10)/Sp(10) . . . . . . . . . . . . . . 57
3.2 TheWeightMonoidsofSL(7)/S(GL(4)×GL(3))andSL(8)/S(GL(4)×
GL(4)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 Weight Monoids Generated by Sets of Fundamental Weights
of G = SL(n+1) . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Faces of the Dominant Cone of SL(n) 73
4.1 Weight Monoids Generated by Fundamental Weights . . . . . 73
vi
CONTENTS vii
4.1.1 Some facts about ΣN(Γ) . . . . . . . . . . . . . . . . . 78
4.1.2 Proof of Proposition 4.9.(1) and 4.9.(2) . . . . . . . . 82
4.1.3 Proof of Proposition 4.9.(3) . . . . . . . . . . . . . . . 100
4.2 Smooth Weight Monoids of Full Rank for SL(n) . . . . . . . . 105
Bibliography 112
Introduction
Let G be a connected reductive algebraic group over C, in which we fix a
Borel subgroup B and a closed subgroup H ⊂ G. The homogeneous space
G/H (or the subgroup H) is spherical if B acts on G/H with an open orbit.
Examples include flag varieties (where H is parabolic in G) and symmetric
spaces (where H is the fixed point set of an involutive automorphism of G).
A large part of the interest in spherical subgroups comes from represen-
tation theory (Gel’fand pairs, multiplicity-free spaces). Indeed, the homoge-
neousspaceG/H is sphericalifandonlyifforanysimple, rationalG-module
M, and for any multiplicative character χ of H, the χ-eigenspace of H in M
is zero or a line.
More generally, one defines a spherical variety as a normal algebraic
variety with an action of G and a dense orbit of B. In the case where G is a
torus(i.e. G = (C∗)k forsomek),thisrecoversthedefinitionofatoricvariety.
The geometry of spherical varieties combines that of flag varieties (e.g. the
Bruhat decomposition and the Borel-Weil-Bott theorem) and of symmetric
spaces (e.g. the little Weyl group and its role in equivariant embeddings).
Formorebackgroundinformationonsphericalvarietieswereferto[Bri94]
inthe1994ICMproceedings,fromwhichtheabovewasadapted,andto[Tim11].
Before giving a brief overview of the thesis, we introduce some notation
andprovidesomebackgroundinformation. TheweightlatticeofGisdenoted
by Λ. It is the character group of T and can be identified with the character
group of B. The set of dominant weights of G with respect to B will be
denoted by Λ+. It is a finitely generated submonoid of Λ. Highest weight
theory says that the irreducible representations of G are classified by the
elements of Λ+. We denote by V(λ) the irreducible G-module corresponding
to λ ∈ Λ+.
1
INTRODUCTION 2
An affine G-variety X is spherical if and only if it is normal and the ring
C[X] of regular functions on X is multiplicity-free as a representation of G.
A combinatorial invariant of such a variety is its weight monoid Γ(X): it is
the set of isomorphism classes of irreducible representations of G that occur
in the coordinate ring C[X] of X. We identify Γ(X) with a finitely generated
submonoid of the monoid Λ+ of dominant weight:
Γ(X) = {λ ∈ Λ+ : Hom (V(λ),C[X]) (cid:54)= 0}
G
It is possible for two non-isomorphic affine spherical varieties to have
the same weight monoid, but the weight monoid is a complete invariant
for smooth affine spherical varieties. Indeed, in the mid 1990s, F. Knop
conjectured and in [Los09a] I. Losev proved the following result:
Theorem. A smooth affine spherical G-variety X is uniquely determined
(up to equivariant isomorphism) by its weight monoid Γ(X).
One of the motivations to study smooth affine spherical G-varieties is
an application to Hamiltonian K-manifolds where K ⊂ G is a maximal
compact subgroup. These are symplectic K-manifolds which are equipped
with a moment map. Locally, a Hamiltonian K-manifold is isomorphic to
a smooth affine G-variety [Sja98]. A Hamiltonian K-manifold is called mul-
tiplicity free if all symplectic reductions are zero-dimensional. Multiplicity
free Hamiltonian K-manifolds have smooth affine spherical varieties as local
models [Bri86]. In [Kno11], Knop proved the following two facts:
(a) A multiplicity-free Hamiltonian manifold for a connected compact Lie
Group is uniquely determined by its generic isotropy group and its mo-
ment polytope.
(b) Locally, the moment polytope of a multiplicity-free Hamiltonian mani-
fold looks like the weight monoid of a smooth affine spherical variety.
Part (a) was known as Delzant’s conjecture [Delzant90] and Knop used
the theorem above to prove it.
The image of the map X (cid:55)→ Γ(X) that sends a smooth affine spherical
variety to its weight monoid is still somewhat mysterious. We call a sub-
monoid Γ of Λ+ smooth if it lies in the image of this map, that is, if there