MATH 47A FALL 2008
INTRO TO MATH RESEARCH
KIYOSHI IGUSA
Contents
1. Permutations or: group theory in 15 minutes
Date : September 3, 2008.
0
1
MATH 47A FALL 2008
INTRO TO MATH RESEARCH
1
1. Permutations or: group theory in 15 minutes
For those of you who a
38
NOTES
5.5.5. n = 5. Today we looked at noncrossing partitions of n = 5. We
expect to get:
C (5) = 42
noncrossing partitions. We got 10 patterns:
(1) 1 with one part
'$
&%
(2) 2 patterns with 2 parts. Each gave 5 ncps for 2 5 = 10 ncps:
'$
'$
&%
&%
(3)
36
NOTES
5.5. noncrossing partitions. This is a partition of the set cfw_1, 2 , n
into parts so that the parts dont cross when you draw the points in a
circle and draw a convex curve surrounding each part.
Here is the plan of how to study any new concept:
MATH 47A FALL 2008
INTRO TO MATH RESEARCH
15
4. Roots and reflections
4.1. ro ots of typ e An1 .
Denition 4.1. The roots of type An1 are the vectors in Rn of the
form
ei ej
First I had to explain to students what Rn means. The standard
notation is:
R = th
10
NOTES
3.3. conjugation.
Denition 3.5. We say that a is conjugate to b if there exists a c so
that
a = cbc1
I used the following notation. First, a b for a is conjugate to b and
c (b) := cbc1
The function c is a homomorphism (a concept that I will expla
MATH 47A FALL 2008
INTRO TO MATH RESEARCH
7
3. more group theory
Today we talked about the main properties of the elements of a
group (although I didnt dene a group yet). I just said that these are
all permutations. I changed the format of the class to pr
MATH 47A FALL 2008
INTRO TO MATH RESEARCH
33
But the original Dyck path is LAB R. So, LA is a beginning of the
original Dyck path. By assumption, this word has more Ls than Rs.
So, if we remove one L we have at least as many Ls as Rs. So, the rst
part A o
MATH 47A FALL 2008
INTRO TO MATH RESEARCH
31
5.3. Dyck paths to binary trees. Last time we constructed Dyck
paths out of binary trees. This can be described in set theoretic language as follows. We constructed a mapping
f : cfw_rooted binary trees with n
MATH 47A FALL 2008
INTRO TO MATH RESEARCH
27
5.2. Dyck paths. First we discussed random walks. This is a stochastic process which you learn about in Math 56a. Then we converted
this into a Dyck paths. Then I showed the bijection from binary trees to
Dyck
4
NOTES
2. Transpositions
On Day 2, I tried to formalize things we talked about at the end
of Day 1, namely, in what way do the crossings in the diagram give
transpositions and: What is the longest word ?
Denition 2.1. A transposition is a 2-cycle (ab). F
MATH 47A FALL 2008
INTRO TO MATH RESEARCH
1
1. Permutations or: group theory in 15 minutes
For those of you who already took a course in group theory, you
probably learned about abstract groups which are sets with binary
operations satisfying a list of co
MATH 47A FALL 2008
INTRO TO MATH RESEARCH
41
7. Categories
The categorication of cluster combinatorics began in the 21st
century. Whereas the rst results about clusters were published by
Fuss in 1791. We will look at this new idea and try to understand it