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E-raamat: Spectra of Symmetrized Shuffling Operators

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Spectra of Symmetrized Shuffling OperatorsSuurem pilt
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The treatise grew out of a desire to understand why a certain family of combinatorial matrices were pair-wise commuting and had only integer eigenvalues. It defines and studies a set of operators at various levels of generality: hyperplane arrangements, hyperplane arrangements invariant under a (linear) action of a finite group, reflection arrangements corresponding to a real reflective group, crystallographic reflection groups (or, equivalently, Weyl groups), and a symmetric group. The general themes are the original family of matrices, using the W-action, an eigenvalue integrality principle, and a broader context with more surprises. Annotation ©2014 Ringgold, Inc., Portland, OR (protoview.com)
Chapter I Introduction
1(6)
1 The original family of matrices
1(1)
2 Using the W-action
2(1)
3 An eigenvalue integrality principle
2(1)
4 A broader context, with more surprises
3(2)
5 Outline of the paper
5(2)
Chapter II Denning the operators
7(16)
1 Hyperplane arrangements and definition of vo
7(2)
2 Semidefiniteness
9(1)
3 Equivariant setting
9(1)
4 Z2-action and inversions versus noninversions
10(1)
5 Real reflection groups
10(3)
6 The case where O is a single W-orbit
13(2)
7 A reduction to isotypic components
15(3)
8 Perron-Frobenius and primitivity
18(5)
Chapter III The case where O contains only hyperplanes
23(14)
1 Review of twisted Gelfand pairs
23(1)
2 A new twisted Gelfand pair
24(2)
3 Two proofs of Theorem I.4.1
26(3)
4 The eigenvalues and eigenspace representations
29(5)
5 Relation to linear ordering polytopes
34(3)
Chapter IV Equivariant theory of BHR random walks
37(22)
1 The face semigroup
37(1)
2 The case relevant for $vo
38(2)
3 Some non-equivariant BHR theory
40(3)
4 Equivariant structure of eigenspaces
43(5)
5 (W x Z2)-equivariant eigenvalue filtration
48(3)
6 Consequences for the kernels
51(1)
7 Reformulation of WHox
52(7)
Chapter V The family V(2k,1n-2k)
59(6)
1 A Gelfand model for n
59(1)
2 Proof of Theorem I.4.3
60(5)
Chapter VI The original family V(k,1n--k)
65(26)
1 Proof of Theorem I.1.1
65(4)
2 The kernel filtration and block-diagonalization
69(2)
3 The (unsigned) maps on injective words
71(1)
4 The complex of injective words
71(1)
5 Pieri formulae for Gn and Gn x Z2
72(2)
6 Some derangement numerology
74(2)
7 (Gn x Z2)-structure of the first kernel
76(2)
8 (Gn x Z2)-structure of the kernel filtration
78(2)
9 Desarrangements and the random-to-top eigenvalue of a tableaux
80(2)
10 Shaving tableaux
82(3)
11 Fixing a small value of k and letting n grow
85(2)
12 The representation X(n-1,1)
87(4)
Chapter VII Acknowledgements
91(2)
Appendix A Gn-module decomposition of V{k,1n--k) 93(6)
Bibliography 99(4)
List of Symbols 103(4)
Index 107
Victor Reiner, University of Minnesota, Minneapolis, Minnesota.

Franco Saliola, Universite du Quebec a Montreal, Canada.

Volkmar Welker, Philipps-Universitaet Marburg, Germany.
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