Total Positivity and Network Parametrizations: From Type A to Type C

126 Pages·2016·0.786 MB·English

by  Rachel Karpman

Total Positivity and Network Parametrizations:
From Type A to Type C
by
Rachel Karpman
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Mathematics)
in The University of Michigan
2016
Doctoral Committee:
Professor Thomas Lam, Chair
Associate Professor Henriette D-M Elvang
Professor Sergey Fomin
Associate Professor David E. Speyer
Professor John R. Stembridge
Rachel Karpman 2016
(cid:13)c
All Rights Reserved
Dedicated to my mother, Constance Medora Dinges. May her memory be a blessing.
ii
ACKNOWLEDGEMENTS
I would like to thank my advisor Thomas Lam for his support throughout the
dissertation process. I am grateful to Thomas for introducing me to algebraic combi-
natorics; for spending many hours with me discussing mathematics; and for sharing
his conjecture about bridge graphs, which became the basis for much of my thesis.
Most of all, I am grateful for his unwavering confidence in me and in the work.
I would like to thank the remaining members of my thesis committee, David
Speyer, Henriette Elvang, Sergey Fomin, and John Stembridge, for their thoughtful
feedback and incisive questions. For her wise and compassionate advice on all as-
pects of graduate school, I am indebted to Karen Smith. Nina White has been a
wonderfulteachingrolemodel, andinspiredmetobeamoreengagedinstructor. The
administrative staff of the mathematics department have all been incredibly helpful.
I am especially grateful to Tara McQueen, Stephanie Carroll and Chad Gorski.
While in graduate school, I have benefited from discussing mathematics with a
distinguished group of faculty and fellow students. I would like to thank Shifra Reif
for her ongoing mathematical guidance, and Lauren Williams for sharing invaluable
insights during my visit to Berkeley. I am grateful to Jake Levinson, Gabriel Frieden,
Yi Su and Timothy Olson for many helpful conversations; and to David Speyer and
Greg Muller for sharing their manuscript [25].
I would like to thank my family, Maurice Karpman, Shira Horowitz, Gabriel
Karpman, and Asher Karpman, for their boundless love and support. I am grateful
iii
to Maurice, Shira and Asher for traveling all the way to Michigan to attend my
dissertation defense, and to Gabriel for constantly inspiring me with his brilliance
and integrity.
My friends at the University of Michigan have sustained me with laughter and
commiseration, warmth and solidarity, gin and cake. Thanks especially to Becca and
Gene, for being the world’s best housemates; and to Patricia, Jake, Ashwath, Rohini
and Rob, for everything.
iv
TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
CHAPTER
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Totally nonnegative matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Total nonnegativity for reductive groups and flag varieties . . . . . . . . . . 4
1.3 Projected Richardson varieties and total nonnegativity . . . . . . . . . . . . 6
1.4 The positroid stratification of the Grassmannian . . . . . . . . . . . . . . . . 6
1.5 Parametrizing positroid varieties . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.1 The boundary measurement map . . . . . . . . . . . . . . . . . . . 9
1.5.2 Deodhar parametrizations for positroid varieties . . . . . . . . . . . 11
1.6 From subexpressions to networks. . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Extending positroid combinatorics . . . . . . . . . . . . . . . . . . . . . . . . 13
1.8 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.8.1 Bridge graphs and Deodhar parametrizations for positroid varieties 13
1.8.2 Total positivity for the Lagrangian Grassmannian . . . . . . . . . . 15
II. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 Notation for partitions, root systems and Weyl groups. . . . . . . . . . . . . 18
2.2 Flag varieties, Schubert varieties and Richardson varietes . . . . . . . . . . . 20
2.2.1 Type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Type C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Projected Richardson varieties and P-order . . . . . . . . . . . . . . . . . . . 25
2.4 Deodhar parametrizations for projected Richardson varieties . . . . . . . . . 29
2.4.1 Distinguished subexpressions . . . . . . . . . . . . . . . . . . . . . 29
2.4.2 Deodhar’s decomposition . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.3 Total nonnegativity. . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.4 Parametrizing Deodhar components . . . . . . . . . . . . . . . . . 32
2.4.5 A pinning for SL(n) . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.6 A pinning for Sp(2n) . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 Positroid varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Grassmann necklaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.7 Bounded affine permutations and Bruhat intervals in type A . . . . . . . . . 37
2.8 Plabic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.9 Parametrizations from plabic graphs . . . . . . . . . . . . . . . . . . . . . . . 46
v
2.10 Γ-diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
III. Bridge graphs and Deodhar parametrizations . . . . . . . . . . . . . . . . . . 52
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.1 Wiring diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.2 Bridge diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 The Main Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 Rewriting Deodhar parametrizations . . . . . . . . . . . . . . . . . 56
3.2.2 From PDS’s to bridge graphs . . . . . . . . . . . . . . . . . . . . . 58
3.2.3 From bridge graphs to PDS’s . . . . . . . . . . . . . . . . . . . . . 68
3.3 Local moves for bridge diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3.1 Isotopy classes of bridge diagrams . . . . . . . . . . . . . . . . . . . 76
IV. Total positivity for the Lagrangian Grassmannian . . . . . . . . . . . . . . . 80
4.1 Bounded affine permutations and Bruhat intervals in type C . . . . . . . . . 81
4.2 Bridge graphs and Deodhar parametrizations for Λ(2n) . . . . . . . . . . . . 89
4.3 The Lagrangian boundary measurement map . . . . . . . . . . . . . . . . . . 92
4.3.1 Symmetric plabic graphs. . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.2 Local moves for symmetric plabic graphs . . . . . . . . . . . . . . . 97
4.3.3 NetworkparametrizationsforprojectedRichardsonvarietiesinΛ(2n).107
4.4 Total nonnegativity for Λ(2n) . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5 Indexing projected Richardson varieties in Λ(2n) . . . . . . . . . . . . . . . . 113
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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LIST OF FIGURES
Figure
1.1 A weighted planar directed graph. Unlabeled edges have weight 1. The dark edges
show the only nonintersecting family of paths from (u ,u ) to (v ,v ). Notice that
1 2 1 2
this family of paths has weight xyz.. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 A weighted plabic graph. Unlabeled edges have weight 1. . . . . . . . . . . . . . . 9
1.3 A bridge network. All unlabeled edges have weight 1. . . . . . . . . . . . . . . . . . 10
1.4 Constructing a planar network from a Γ-diagram for a positroid cell in Gr (2,5) . 10
≥0
1.5 A bridge network. All unlabeled edges have weight 1. . . . . . . . . . . . . . . . . . 14
1.6 A symmetric weighting of a symmetric plabic graph. Unlabeled edges have weight 1. 17
2.1 Realizing Λ(2n) as a submanifold of Gr(n,2n). . . . . . . . . . . . . . . . . . . . . 25
2.2 Crossings in a chord diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Alignmentsinachorddiagram. Notethatloopsmustbeorientedasshowntogive
a valid alignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 A square move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 A reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.6 Adding bridges to a plabic graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.7 A square move with accompanying change of coordinates. . . . . . . . . . . . . . . 49
2.8 Constructing Γ-diagrams for the case k = 2, n = 5. The diagrams at right corre-
spond to expressions s s s s s and s s 1s 1, respectively . . . . . . . . . . . . . . 50
2 1 4 3 2 2 1 3
2.9 We construct type B Γ-diagrams inside a staircase shape. The figure shows n=3. 51
3.1 Replacing a crossing with a bridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 If the left endpoint of wire b is ≤k, the same must be true for wire a. . . . . . . . 63
3.4 Adding a rightmost bridge (a ,b ) between two non-isolated wires corresponds to
r r
adding a bridge between two adjacent legs of a planar network. . . . . . . . . . . . 64
3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7 Adding a segment q of wire q to the bridge diagram B . Here q corresponds to a
i i
black lollipop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.8 Adding a lollipop to the bridge graph G gives a bridge graph G(cid:48). We construct
a bridge diagram for G(cid:48) by adding a new wire (dashed) to a bridge diagram for
G. The portion of each bridge diagram to the right of the thick vertical line is the
restricted part. The vertical lines divide the restricted part of each diagram into
segments, as in the proof of Lemma III.18. . . . . . . . . . . . . . . . . . . . . . . . 72
3.9 Braid moves for PDS’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.10 B and B are isotopic valid bridge diagrams with u=42153 and w =42531. For
1 2
i=1,2, the portion of B to the right of the dashed line is B0. . . . . . . . . . . . 79
i i
4.1 A symmetric bridge graph with symmetric weights. . . . . . . . . . . . . . . . . . . 90
4.2 A symmetric weighting of a symmetric plabic graph. All unlabeled edges have
weight 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 A symmetric reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4 The distinguished diameter d divides a symmetric plabic graph into two regions,
which correspond to a pair of plabic graphs G and G . . . . . . . . . . . . . . . . 100
1 2
vii
ABSTRACT
Total Positivity and Network Parametrizations: From Type A to Type C
by
Rachel Karpman
Chair: Thomas Lam
The Grassmannian Gr(k,n) of k-planes in n-space has a stratification by positroid
varieties, which arises in the study of total nonnegativity. The positroid stratification
has a rich combinatorial theory, introduced by Postnikov. In the first part of this
thesis, we investigate the relationship between two families of coordinate charts, or
parametrizations, of positroid varieties. One family comes from Postnikov’s theory of
planarnetworks,whiletheotherisdefinedintermsofreducedwordsinthesymmetric
group. We show that these two families of parametrizations are essentially the same.
In the second part of this thesis, we extend positroid combinatorics to the Lagrangian
Grassmannian Λ(2n), a subvariety of Gr(n,2n) whose points correspond to maximal
isotropic subspaces with respect to a symplectic form. Applying our results about
parametrizations of positroid varieties, we construct network parametrizations for
the analogs of positroid varieties in Λ(2n) using planar networks which satisfy a
symmetry condition.
viii
CHAPTER I
Introduction
1.1 Totally nonnegative matrices
Aninvertiblematrixistotally positive ifallofitsminorsarepositiverealnumbers.
For example, the matrix
 
14 7 2
 
 
 
5 3 1
 
 
 
2 2 1
is totally positive. Similarly, an invertible matrix is totally nonnegative if all of its
minorsarenonnegativerealnumbers. Wedenotethegeneral linear group ofinvertible
n × n matrices by GL(n), the subset of totally positive matrices by GL (n), and
>0
the subset of totally nonnegative matrices by GL (n). Note that GL (n) is the
≥0 ≥0
closure of GL (n).
>0
Totally positive matrices first appeared in a paper of Schoenberg in 1930 [32].
A few years later, Gantmacher and Krein showed that all eigenvalues of a totally
positive matrix are simple, real and positive [9, 10]. Total positivity has proved to
be a powerful tool in many areas of mathematics, including analysis, probability, and
applied mathematics [12].
Since the 1980’s, there has been a great deal of research on the interplay between
totalpositivityandcombinatorics[6]. AkeyresultinthisareaisLindstro¨m’sLemma,
1