You will need to know what each of the commands does and Maple will tell you by typing "?" followed by a command name. For example "? subs" gives a page on how the command "subs" (short for 'substitute') works.

A short tutorial on Maple is here.

Notes on symmetric functions that are similar to a computer development.

Or broken into chapters: Chapter 1, Chapter 2, Chapter 3, Chapter 4, Chapter 5.

David Little's notes on partitions and species.

- Introduction to groups and
representations
- This lab develops a way of working with groups, modules,
representations and characters. The purpose of this lab is to
experiment with and
break down several modules into their irreducible submodules. We
will
use some of these functions in the other labs.

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- Decompose the module of the regular representation of S_3 and S_4 into irreducible submodules
- Calculate the conjugacy classes of Dn for n=2,3,4,5,6. Give a description of the conjugacy classes of Dn and then prove your answer.
- Try to find a matrix T such that T &* permrep5(j) &*
T^(-1)
is diagonal for all 1 \leq j < 5. What does this say about the
the
matrix representation?

- The Quotient of C[x1, x2, ..., xn] by
the ideal generated by the symmetric functions - We use techniques
of
Groebner bases to work with the ideal < e_i(x1, x2, ..., xn) :
i=1...
n>. A Grobner basis of an ideal is a minimal set of generators
which are needed for a computer to "work nicely" with an ideal.
Maple
has packages built in to do these these computations. There is a
natural
S_n action on this quotient space and so it can be viewed as a graded
S_n
module. The main exercise of this lab will be to decompose the space
for
n=4.

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- Find a basis for the ring C[x1,x2,x3,x4]/<e1,e2,e3,e4>, grade this basis by degree since the S_4 action will preserve the degree on the elements of the ring.
- Define an S_4 action on the basis you found
- Determine the graded Frobenius image of the ring C[x1, x2,
x3, x4]/<e1,e2,e3,e4>. In this exercise you will break this
module down into components graded by degree and then determine the
Frobenius image.

- The Specht module - In this lab
we construct an irreducible representation of S_n for each partition of
n and we can show that it is irreducible by computing the scalar
product of the character of the module with itself. Finally we
experiment with the branching rule and determine what happens to the
module when we restrict from S_n to S_{n-1}.

HTML version

- Determine which irreducible representations of S_7 when
restricted to S_6 will also be irreducible. How will all of the
irreducible representations of S_7 break up under this restriction?
Will there ever be any irreducible representations of S_n which when
restricted to S_{n-1} break into representations with multiplicity
greater than 1?

- Determine which irreducible representations of S_7 when
restricted to S_6 will also be irreducible. How will all of the
irreducible representations of S_7 break up under this restriction?
Will there ever be any irreducible representations of S_n which when
restricted to S_{n-1} break into representations with multiplicity
greater than 1?

- The Schur functions - This lab
experiments with several definitions of the Schur functions and shows
how they are equivalent. By finding the coefficient of the Schur
function in the power basis the exercise will be to give the character
table for S_n. We will
use these functions in other labs to manipulate symmetric functions.

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- Write the functions "Schur_in_e" and "skew_Schur_in_e" (analogous to "Schur_in_h") using the Jacobi-Trudi identity for the e-basis.
- Use the identity P(t) = ln( H(t) ) to write the functions "pn2h" and "p2h" modeled after the functions "hn2p" and "h2p." Use these functions to show that p2h(e2p(Schur_in_e(la))) = Schur_in_h(la) for various paritions la.
- verify the correspondence between each of the monomials in the Schur function s[2,2,1](x1,x2,x3,x4) and the column strict tableaux of shape [2,2,1] in the values 1,2,3,4.
- What happens when we multiply a Schur function by hk, pk or
ek?
Calculate the examples and explain the computer responses,

- In the Jacobi-Trudi defintion, it is not necessary that the
indexing sequence be a partition. Our function ehp2s will expand
this in the Schur basis (which is indexed by a partition). What
happens?

- Counting necklaces - We will count the
number of ways of coloring a necklace with k beads and n colors in two
ways, one of them will be to use symmetric functions the other will be
to create lists of numbers which represent necklaces and count the
number of orbits under the action of a group.

HTML version

- count the number of different necklaces with 5 beads with 3 colors if the beads are allowed to rotate around the necklace
- count the number of different necklaces with 5 beads and 3 colors if the beads are allowed to rotate around the necklace or the necklace can be flipped over
- count the number of different necklaces with 5 beads with 3 red, and 1 blue and 1 green if the beads are allowed to rotate around the necklace
- count the number of different necklaces with 5 beads with 3 red, and 1 blue and 1 green if the beads are allowed to rotate around the necklace or it can be flipped over.
- Count the number of ways of coloring the faces of the cube
with 6 colors so that all 6 faces are a different color.

- Tableaux and symmetric functions -
In this lab we use the Robinson-Schenstead-Knuth algorithm to decompose
symmetric functions in the Schur basis.

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- Add all Schur_in_mon(T, 5) for T all standard tableaux of shape 3 and factor the result (next do 4 and 5). What do you see? Why? How does this show that sum_la f_la s_la = h1^n?
- Let S be the set of sequences of length 6 in the numbers
1,2,3,4 that may have descents in positions 3 and 4. Show that
the sum over x[w[1]]*x[w[2]]*...*x[w[6]] for w in S is a homogeneous
symmetric function.

Convert each w in S to a pair of tableau and select out the "leading terms" of the Schur functions and then expand these Schur functions in the h-basis. Do you get the same answer if the words only have the numbers 1,2,3? What happens if you only have the numbers 1 and 2 in the words?