Lab10-Representation-theory.sagews

Representations of the symmetric group

Math 737 - Lab 10

Exercise 1: For $n=3,4,5,6$, use the function '.irreducible_characters()' to show that the number of irreducible characters of $S_n$ is the number of partitions of $n$.

S3 = SymmetricGroup(3)
irr = S3.irreducible_characters()
len(irr)

Exercise 2: The code below does the following for $n=3$. Do the same for $n=4,5,6$.

• Compute the character table of $S_n$ (including the conjugacy class representatives indexing the columns).

• By looking at the output and using your knowledge of character tables, state which partition corresponds to each row.

S3 = SymmetricGroup(4)
S3.conjugacy_classes_representatives()
S3.character_table()
print "The partitions indexing the rows (in order) are:",[1,1,1],[2,1],[3]

[(), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4)] [ 1 -1 1 1 -1] [ 3 -1 -1 0 1] [ 2 0 2 -1 0] [ 3 1 -1 0 -1] [ 1 1 1 1 1] The partitions indexing the rows (in order) are: [1, 1, 1] [2, 1] [3]

Exercise 3: The code below does the following for $n=3$. Do the same for $n=4,5,6$.

• Check that the row orthogonality theorem holds.

irr = S3.irreducible_characters()
#irr is the list of irreducible characters. Since it is a list, we can access each of these three elements as irr[0], irr[1], irr[2].
irr[0].values()
irr[1].values()
irr[2].values()

[1, -1, 1] [2, 0, -1] [1, 1, 1]

#Now use .scalar_product to perform the inner product of one character with another.
irr[0].scalar_product(irr[0])
irr[0].scalar_product(irr[1])
irr[0].scalar_product(irr[2])
irr[1].scalar_product(irr[0])
irr[1].scalar_product(irr[1])
irr[1].scalar_product(irr[2])
irr[2].scalar_product(irr[0])
irr[2].scalar_product(irr[1])
irr[2].scalar_product(irr[2])
#Need to check the scalar product for all combinations of irr[i], irr[j] for values of i,j between 0 and 2. You'll want to write a loop for higher n, since there will be many more combinations for S_4, S_5, and S_6.

1 0 0 0 1 0 0 0 1

Exercise 4: The code below does the following for $n=3$. Do the same for $n=4,5,6$.

• Sum the squares of the dimensions of each irreducible representation to obtain the order of the group.

#Here is the calculation for n=3. For higher n, you should write a loop rather than writing the sum manually.
list(irr[0])[0]^2+list(irr[1])[0]^2+list(irr[2])[0]^2

6

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