Math 821, Spring 2013, Lecture 13
Karen Yeats
(Scribe: Mahdieh Malekian)
March 5, 2013
1
SubHopf Algebra Results
1
Example. Last time we showed that T (x) = B+ ( T (x) ).
r
T (x) = r x r x2
r
r
r
r
r
r
r + r r x3 r + r + 2 r r + r r r x4 +O(x5 ).
r
r r
r
Math 821 Combinatorics Notes
Karen Yeats
(Scribe: Avi Kulkarni)
March 21, 2013
1
1.1
Schur functions
Introduction
These are possibly the least natural but most important basis for symmetric functions.
Denition A semistandard Young tableau of shape (or col
Math 821, Spring 2013, Lecture 2
Karen Yeats
(Scribe: Mahdieh Malekian)
January 17, 2013
1
Generating Functions
Last time we dened A = B C, A = B + C, E, Z.
Notation. Let B be a combinatorial class. Write B n = B . . . B, for n > 0,
n times
and B 0 = E.
D
MATH 821, SPRING 2012, ASSIGNMENT 3 SOLUTIONS
(1) First we have a trilinear map
f : V1 V2 V3 V1 (V2 V3 )
given by f (v1 , v2 , v3 ) = v1 (v2 v3 ). Fixing any value v3 in the third component
f (, , v3 ) is bilinear and so by the universal property of tenso
Math 821, Spring 2013
Karen Yeats
(Scribe: Wei Chen)
February, 19, 2013
Algebras
Denition. Let k be a eld. An (associated) algebra A over k is a vector space with a binary operation
satisfying
(1) is associative
(2) distributes over + on the left and rig
MATH 821, SPRING 2013, ASSIGNMENT 4 SOLUTIONS
(1) Given a self conjugate partition of n. Consider the Ferrers diagram of . Take
those boxes which are in the top row or the rst colum (including the top corner
which is in both). Make those boxes the rst row
MATH 821, SPRING 2012, ASSIGNMENT 2 SOLUTIONS
(1) (a) Let C be an unlabelled combinatorial class with C0 = and let B = MSet(C).
Then
(1 x|x| )1
B(x) =
cC
since each element can appear any number of times. Therefore
(1 x|x| )1
B(x) =
cC
(1 xn )cn
=
n=1
(1
Math 821, Spring 2013, Lecture 5
Karen Yeats
(Scribe: Avery Beardmore)
January 29, 2013
1
A Taste of Bivariate Generating Functions
Let P be a combinatorial class of permutations.
P = Set(DCyclabelled (Z) Sequnlabelled (Z)
So
P (x) = exp log
1
1x
=
1
1x
L
Math 821, Spring 2013, Lecture 3
Karen Yeats
(Scribe: Yue Zhao)
January 22, 2013
1
Constructions coming from cycle index polynomial
Recall:
Denition. Let A be a permutation group on cfw_1, 2, . . . , n, Z(A; s1 , s2 , . . . , sn ) =
1
|A|
n
j ()
skk
. Whe
Math 821, Spring 2013, Lecture 15
Karen Yeats
(Scribe: Avery Beardmore)
March 12, 2013
1
Partitions
Denition 1. A partition of n is = (1 . . . k ) such that 1 + . . . +
k = n, where
n is the size of
k is the number of parts
Write k() for the number of
Math 821 Lecture 6
Ross Churchley
January 31
Stirlings approximation
Youve probably seen Stirlings approximation at some point in your education, but you may not have seen (or remember) the proof. For the first
part of today, well prove Stirlings formula1
MATH 821, Spring 2013, Lecture 17
Karen Yeats
(Scribe: TJ Yusun)
March 19, 2013
Outline: Coproduct in (x), (x) is a Hopf algebra, other bases for (x).
1
on Symmetric Functions
The ring of symmetric functions (x) has a coproduct. First, we need to underst
MATH 821, Spring 2013, Lecture 4
Karen Yeats
(Scribe: TJ Yusun)
January 24, 2013
Outline: Labelled combinatorial classes, exponential generating functions, order shufes, the usual constructions, Set and PSet, set partitions and permutations, fake
labelled
Math 821, Spring 2013
Karen Yeats
(Scribe: Wei Chen)
March, 26, 2013
What about the antipode.
Proposition.
(a) S(pn ) = pn
(b) S(en ) = (1)n hn
(c) S(hn ) = (1)n en
Recall the fundamental involution :
Proof of Proposition.
(a) From the fact that pn are p
Math 821, Spring 2013, Lecture 12
Karen Yeats
(Scribe: Yue Zhao)
February 28, 2013
1
Duals
Denition. (1) Let V be a nite dimensional vector space over k, then
V = Hom(V, k) the space of linear maps from V to k.
(2) If : V W is a linear map, then : W V by
Math 821 Lecture Notes
Ross Churchley
March 14 (Happy Pi Day!)
Jacobi triple product formula
Proposition 1.
2
xh yh =
h=
(1 x2i1 y )(1 + x2i1 y 1 )(1 x2i )
i =1
Proof. It suffices to show
1
1 x2i
i =1
2
xh yh =
h=
(1 x2i1 y )(1 + x2i1 y 1 ).
(1)
i =1
The
Math 821, Spring 2013, Lecture 8
Karen Yeats
(scribe: Yian Xu)
February 7, 2013
1
A Result
Denation 1 Let T (x) be a formal power series. Dene Supp(T (x)
to be the set of the indices of nonzero coecients of T (x). Dene
d = min(Supp(T (x), g = gcd(Supp(T (