MATH 581 Problem Set 2. Due Tuesday February 19.
Problem 1. Let be a partition with 1 m. Use the Gessel-Viennot Theorem on a appropriate
graph to show that
s = det ei i+j 1i,j m .
Problem 2. For any m, n N, consider the determinant
D(m, n) = det
n
i+j
.
0
MATH 581 Problem Set 1. Due Thursday January 24.
Problem 1. For any f n , dene fk nk by
fk (x1 , x2 , . . .) = f (xk , xk , . . .).
1
2
Show that
fk = (1)n(k1) (f )k .
Problem 2.
(a) Show that P (t) = E (t)/E (t).
(b) Use the fact that (H (t) = E (t) to s
3
The HHL formula
References. [HHL05] [HHL08]
Assume all diagrams are in the French notation (diagrams live in the rst quadrant, as opposed to the fourth quadrant for English notation).
A lling of a partition ` n is a map : ! N. Write
Y
x=
x (u) .
u2
Giv
2
All the avours of Macdonald polynomials
Let K = C(q, t) be the eld of rational functions in two parameters q and t.
Denition 16. Let
(x, y; q, t) =
YYY 1
i 1 j 1k 0
1
q k txi yj
.
q k xi yj
At q = 0, the only terms which survive are when k = 0. So this
Math 581, Notes Part II
Martha Yip
April 9, 2013
1
Hall-Littlewood polynomials
References. [Macdonald95, Ch. III] [LS]
These polynomials rst arose in Halls study of the Hall algebra for abelian groups,
where he gave an indirect denition for these polynomi
2
The combinatorics of p-cores
We examine some enumerative results on counting cores with respect to their largest
part.
Denition 87. Fix a positive integer ` 2. Given a Ferrers diagram for a partition
, let hk(i, j ) be the hooklength of the box in row i
Math 581, Notes Part II
Martha Yip
March 27, 2013
The goals of Part II of this course are to see some connections between symmetric
functions, combinatorics, and representation theory, and hopefully answer questions
such as: Why compute transitions betwee
6
The Littlewood-Richardson Rule
References. [Stanley, 7.10, 7.17, A1.2, A1.3]
6.1
Skew Schur functions and orthogonality
Denition 131. Given partitions , , , let
so that s s =
P
c = hs , s s i,
c s . Dene the skew Schur symmetric function by
X
s / =
c s
5
Quasisymmetric functions
References. [Stanley, 7.19] [Bergeron, 4.9-4.10]
The quasisymmetric functions were dened by Gessel in the 1980s, who developed
them for the purposes of permutation enumeration. For now, we are interested in these
for their appli
4
Applications of RSK
References. [Stanley, 7.12-7.13, 7.20-7.22]
4.1
Some tableau formulas
Corollary 77. Let , ` n. Then
X
K K
= N ,
`n
where K is a Kostka number and N is the number of matrices with nonnegative
integer entries with row sum and column su
2
Schur symmetric functions
Reference. [Stanley, 7.10, 7.15], [Macdonald95, I.3]
We begin by giving two denitions for Schur functions; one algebraic, and one
combinatorial.
2.1
The algebraic denition
In this section, we work in the ring Z[x1 , . . . , xn
1
Introduction to Symmetric Functions
Reference. [Stanley, 7.1, 7.3 - 7.7] [Macdonald95, I]
1.1
The ring of symmetric functions
Denition 9. Let Z[x1 , . . . , xn ] denote the ring of polynomials over Z. The symmetric
group Sn acts on Z[x1 , . . . , xn ] b
Math 581, Winter 2013 Notes
Martha Yip
January 10, 2013
Abstract
These Notes serve as an abbreviated record of the Lectures. The goals for
the Symmetric Functions part of the course are:
1. Find bases for the ring of symmetric functions and formulas for t
MATH 581 Problem Set 3. Due Tuesday March 19.
Problem 1. Let g (n) be the number of symmetric plane partitions = (ij ) with
whose main diagonal is (11 , 22 , . . .) = . Show that
ij = n,
g (n)xn = s (x, x3 , x5 , . . .).
n0
Problem 2. Suppose
n and () =