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Hyperplane Arrangement Face Algebras
and their Associated Markov Chains
by
T. Patrick Bidigare
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Mathematics)
in The University of Michigan
1997
Doctoral Committee:
Professor Philip Hanlon, Chair
Professor Andreas Blass
Professor Yuri Gurevich
Professor Gopal Prasad
Professor Jeffrey Rauch
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UMI Number: 9721946
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ACKNOWLEDGEMENTS
I thank my advisor, Phil Hanlon, for the time he has devoted to my mathematical
development. His direction and suggestions have led to many of the results in this
thesis. I also appreciate the careful reading and numerous suggestions given by
Andreas Blass.
Over the years, David James, John Fink and A1 Taylor have nurtured my interest
in mathematics and axe largely responsible for my decision to pursue it in graduate
school.
I am grateful for the many friends I have had through graduate school with whom
I have roomed, lifted, sparred and engaged in all types of buffoonery.
Finally, I thank Gram, Lum, Liver and Sarah for their love, prayers and encour
agement.
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TABLE OF CONTENTS
ACKNOW LEDGEM ENTS................................................................................. ii
LIST OF TABLES..................................................................................................... v
LIST OF FIG U RES................................................................................................. vi
CHAPTER
I. Introduction & Sum m ary..................................................................... 1
H. The Face A lgebra................................................................................... 10
2.1 Hyperplane Arrangements............................................................. 10
2.2 Edges and Faces.............................................................................. 12
2.3 Face Algebra Structure................................................................. 21
2.4 Representations of the Face Algebra.......................................... 30
2.5 Eigenvalues of the Chamber Representation............................. 38
HI. Reflection Arrangements........................................................................ 53
3.1 Root Systems & Finite Reflection Groups................................ 54
3.2 Reflection Arrangements................................................................. 58
3.3 Multiplication Form ula................................................................ 65
3.4 Reflection Arrangement Eigenvalues.......................................... 68
3.5 Isotypic Subspaces.......................................................................... 70
3.6 The Trivial Isotypic Subalgebra................................................... 74
3.7 Nilradical of the Trivial Isotypic Subalgebra............................. 78
3.8 Solomon’s Descent Algebra.......................................................... 85
IV. Combinatorial Interpretations........................................................... 90
4.1 The Symmetric Group.................................................................... 91
4.2 The Hyperoctahedral Group ..........................................................112
V. Markov Chain Applications.....................................................................130
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5.1 Markov Chain Basics .....................................................................130
5.2 Maxkov Chains on Permutations of Objects..............................135
5.3 Maxkov Chains on Other Types of Objects ..............................144
BIBLIOGRAPHY.......................................................................................................149
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LIST OF TABLES
Table
2.1 Basis Vectors for Edge Space..................................................................... 44
4.1 Face/BOP Correspondences for A-i ......................................................... 99
4.2 Symmetric Face Algebra Dimensions .........................................................106
4.3 Symmetric Face Algebra Eigenvalues........................................................107
4.4 Symmetric Descent Algebra Antiisomorphism.........................................108
4.5 Bases for the Ring of Symmetric Functions...............................................109
4.6 Eigenvalues where y. = cycle — type(w)...............................................Ill
4.7 Edge/BBS Correspondences for B ............................................................120
2
4.8 Face/BOSS Correspondences for B2 ............................................................122
4.9 Hyperoctahedral Face Algebra Dimensions...............................................128
4.10 Hyperoctahedral Eigenvalues........................................................................129
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LIST OF FIGURES
Figure
1.1 Tsetlin Library Transition Diagram......................................................... 1
1.2 Riffle Shuffling Transition D iagram ......................................................... 2
1.3 Data Transmission Model Transition Diagram ..................................... 2
1.4 Some Dimension 3 Hyperplane Arrangements........................................ 3
1.5 Faces of a Hyperplane Arrangement......................................................... 4
1.6 The Action of Faces on Chambers............................................................ 4
1.7 Face Shuffling Transition Diagram............................................................ 4
1.8 Face Weights for the Tsetlin Library and Riffle Shuffling.................. 5
1.9 Face Weights for Data Transmission Model............................................ 6
2.1 Boolean Arrangements................................................................................ 11
2.2 Projections of Braid Arrangements......................................................... 12
2.3 Rank 3 Boolean Arrangement Intersection Lattice............................... 15
2.4 Rank 3 Braid Arrangement Intersection L attice.................................. 15
2.5 Faces of the Rank 2 Boolean and Braid Arrangements........................ 17
2.6 Action of Faces on Chambers .................................................................. 34
2.7 Faces of the Rank 2 Boolean Arrangement............................................ 35
2.8 Cone Containment...................................................................................... 42
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2.9 Chambers and Edges for the Rank 2 Boolean Arrangement................ 43
3.1 Rank 3 Reflection Arrangements.............................................................. 56
4.1 Projections of Braid Arrangements........................................................ 93
4.2 Chamber/Permutation Correspondences for A2 ...................................... 94
4.3 Edge/Partition Correspondences for A2 ................................................. 95
4.4 Faces of A2...................................................................................................... 98
4.5 Hyperoctahedral Reflection Arrangements..............................................115
4.6 Chamber/Permutation Correspondences forZ ?2 .........................................117
4.7 Faces of Z?2.........................................................................................................122
5.1 Face Shuffling Transition Diagram..............................................................133
5.2 3-Shuffling Transition Diagram.....................................................................141
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