A q-analogue of spanning trees : nilpotent transformations over finite fields [PhD thesis]

67 Pages·2009·0.383 MB·English

by  Jingbin Yin

A q-analogue of Spanning Trees: Nilpotent
Transformations over Finite Fields
by
Jingbin Yin
B.S., Peking University, China, 2005
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2009
(cid:13)c Jingbin Yin, MMIX. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part in any medium now known or hereafter created.
Author ..............................................................
Department of Mathematics
April 13, 2009
Certified by..........................................................
Richard P. Stanley
Levinson Professor of Applied Mathematics
Thesis Supervisor
Accepted by.........................................................
Michel X. Goemans
Chairman, Applied Mathematics Committee
Accepted by.........................................................
David Jerison
Chairman, Department Committee on Graduate Students
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A q-analogue of Spanning Trees: Nilpotent Transformations
over Finite Fields
by
Jingbin Yin
Submitted to the Department of Mathematics
on April 13, 2009, in partial fulfillment of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
Abstract
The main result of this work is a q-analogue relationship between nilpotent trans-
formations and spanning trees. For example, nilpotent endomorphisms on an n-
dimensional vector space over F is a q-analogue of rooted spanning trees of the
q
complete graph K . This relationship is based on two similar bijective proofs to
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calculate the number of spanning trees and nilpotent transformations, respectively.
We also discuss more details about this bijection in the cases of complete graphs,
complete bipartite graphs, and cycles. It gives some refinements of the q-analogue
relationship. As a corollary, we find the total number of nilpotent transformations
with some restrictions on Jordan block sizes.
Thesis Supervisor: Richard P. Stanley
Title: Levinson Professor of Applied Mathematics
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Acknowledgments
First and foremost, I would like to express my sincere gratitude to my advisor, Pro-
fessor Richard Stanley, for his advising of my work. His continuous guidance and
insightful comments helped me a lot throughout my Ph.D. study. I would like to
also thank Professor Ira Gessel and Professor Alexander Postnikov for serving on my
thesis examination committee. I owe thanks to the faculty, to my colleagues, and to
the staff of the Department of Mathematices at the Massachusetts Institute of Tech-
nology. They have made this department a friendly and warm community. Finally,
I want to thank my family for their love and support, especially my parents and my
boyfriend.
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Contents
1 Introduction 9
2 Definitions and Main theorems 13
2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Digraphs and Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Proofs 21
3.1 Lemma 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Lemma 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 “Adapt” a Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Examples 35
4.1 Complete Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.1 Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.2 Nilpotent Transformations . . . . . . . . . . . . . . . . . . . . 42
4.2 Complete Bipartite Graphs . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.1 Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.2 Nilpotent Transformations . . . . . . . . . . . . . . . . . . . . 54
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4.3 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.1 Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.2 Nilpotent Transformations . . . . . . . . . . . . . . . . . . . . 63
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Chapter 1
Introduction
Theproblemofenumeratingthenumberofnilpotentmatriceswithcertainrestrictions
over a finite field has attracted great attention in the literature.
In 1958, N.J. Fine, I.N. Herstein in [2], for the first time, found out that the
number of nilpotent n × n matrices over the q-element finite field F is qn(n−1), by
q
considering the decomposition according to Jordan canonical form. Later in 1961, M.
Gerstenhaber gave another proof of it suggested by algebraic geometry in [3]. That
was not the end of the story. After several years, in 1987, A. Kovacs considered the
problem of when the product of k n×n matrices will be nilpotent, and gave a solution
in [5] and [6]. There are more related results.
In general, people are interested in problems of the following form:
Problem. Consider all nilpotent matrices over F of a fixed size with some entries
q
set to be zero. How many are they?
For instance, the problem of when the product of k m×m matrices, A A ···A ,
1 2 k
will be nilpotent is equivalent to consider when the following block matrix A is nilpo-
tent:
 
0 A 0 ··· 0 0
1
 
 0 0 A ··· 0 0 
 2 
 . . . . . . 
A =  .. .. .. .. .. .. .
 
 
 
0 0 0 ··· 0 A
 k−1
 
A 0 0 ··· 0 0
k
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Let’s start the discussion with some examples first.
Bad Example. Consider nilpotent 3×3 matrices of the following form:
 
∗ 0 ∗
 
A = 0 ∗ ∗,
 
 
∗ ∗ 0
where∗denotesaentryfromF . Onecaneasilycomputethetotalnumberofnilpotent
q
matrices over F of the above form is:
q

2q3 −2q2 +2q −1, if q is odd,
q3 +q2 −1, if q is even.
It is not a polynomial of q.
OK Example. Consider nilpotent (n+1)×(n+1) matrices of the following form:
 
∗ ··· ∗ ∗
. . . .
.. .. .. ..
 
A =  ,
 
∗ ··· ∗ ∗
 
 
∗ ··· ∗ 0
where ∗ denotes a entry from F . Define M to be the set of those matrices, i.e., all
q n
nilpotent (n+1)×(n+1) matrices A = (a ) over F such that a = 0. We
i,j q n+1,n+1
want to calculate the cardinality of M .
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Let A (resp. B) be the subset of M such that the last row of any matrix in A
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(resp. in B) is nonzero (resp. zero). We have M is the disjoint union of A and B.
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Let C (resp. D) denote the set of all nilpotent (n+1)×(n+1) matrices A = (a )
i,j
overF suchthat(a ,a ,...,a )isnotazerovector(resp. isazerovector).
q n+1,1 n+1,2 n+1,n
Hence, the set of all nilpotent (n+1)×(n+1) matrices over F is a disjoint union
q
of C and D. We have:
#C +#D = qn(n+1),
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