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189-570A: Higher Algebra I

Assignment 3

Due: Monday, October 16.




I have decided to divide the class into two groups, group A and group B. Group A is for the people who have a somewhat stronger background and are not struggling too much with the assignments. Group B is for the students with somewhat less preparation.
Indicate clearly at the start of your assignment what group you place yourself in.
If you place yourself in group B, you are required to do only problems 1 through 5. However, at the end of the term there will be a price to pay for this lighter assignment load: most likely I will ask you to write a project or a take-home assignment developping in more depth some aspect of the course. The idea, for people in Group B, is to spread the work out more evenly over the semeser so that you are not handicapped by your less strong preparation.


1. Recall that for an element g of a group G, the centralizer of g in G - the group of elements of G which commute with g - is denoted by Z(g).

a. Show that the function which to g in G associates the cardinality of Z(g) is the character of a (naturally occuring) representation of G. What is this representation?

b. Using part a, compute the average over G of the cardinality of Z(g) -- the average size of the centralizer of an element of G.


2. Let G = { 1, -1, i, -i, j, -j, k, -k} be the quaternion group of order 8. (Recall that the multiplication of quaternions is defined by the rules ij = -ji = k, jk = -kj = i, ki = -ik = j.)

a. Show that G is a non-abelian group which is not isomorphic to the dihedral group of order 8.

b. Compute the character table of G and compare it with the character table for D8 computed in class. What do you observe?


3. Compute the character table for S5, the symmetric group on 5 elements.


4. Let V be a representation for G, and let W=V*V be the tensor product of V with itself.

a. The symmetric square of V, denoted Sym2(V), is the subspace of W spanned by expressions of the form v1*v2+v2*v1. Show that the character X attached to Sym2(V) is given by the formula
X(g) = (chi(g)2+chi(g2))/2
where chi denotes the character attached to V.

b. The alternating square of V, denoted Alt2(V), is the subspace of W spanned by expressions of the form v1*v2-v2*v1. Show that the character X attached to Sym2(V) is given by the formula
X(g) = (chi(g)2-chi(g2))/2
where chi denotes the character attached to V.


Let G = GL3(F2) be the group of invertible 3 x 3 matrices with entries in the field with two elements. It acts naturally by linear transformations on the space W of column vectors of size 3 with entries in F2.

The goal of the following series of exercises is to partially calculate the character table for G.

Note that you are not obliged to follow the hints.


5. Show that G has a unique conjugacy class (to be denoted 2A) of elements of order 2, which is of cardinality 21. (Hint: If T is an element of order 2 in G, show that T has minimal polynomial (T-1)2. Conclude that the linear endomorphism U=(T-1) has a two dimensional kernel and that the image of U is a one-dimensional subspace of Ker(U). Show that the datum (Image(U) , kernel(U)) determines T completely.)


6. Show that G has a unique conjugacy class (to be denoted 4A) of elements of order 4, which is of cardinality 42. (Hint: Show that the 21 distinct Sylow 2-subgoups of G are isomorphic to D8 and that the intersection of any two of them is a group of exponent 2.)


7. Show that G has a unique conjugacy class (to be denoted 3A) of elements of order 3, which is of cardinality 56. (Hint: Show that the characteristic polynomial of an element T of order 3 is equal to (x+1)(x2+x+1) and that T is completely determined by the following data: the kernel W1 of T2+T+1 and the kernel W2 of T+1, yielding a decomposition of W into a direct sum of a two-dimensional and a one dimensional subspace, together with the extra datum of a cyclic permutation of order 3 on the non-zero vectors in W1.)


8. Show that G has two distinct conjugacy classes of elements of order 7 (to be denoted 7A and 7B) of elements of order 7, corresponding to the linear transformations having characteristic polynomial x3+x2+1 and x3+x+1 respectively. (Hint: Show that G has 8 distinct Sylow 7-subgroups and hence 48 elements of order 7.)


9. Let X be the set of non-zero vectors of W and let V2=fct0(X,C) be the set of complex-valued functions on X of sum 0. Compute the character attached to V and show that V is an irreducible 6-dimensional representation of G.


10. Let X be the set of Sylow 7-subgroups of G, on which G acts by conjugation, and let V3=fct0(X,C) be the set of complex-valued functions on X of sum 0. Compute the character attached to V and show that V is an irreducible 7-dimensional representation of G.


11. Let W4 be the alternating square of the representation V2 of exercise 9. Show that V4 decomposes as a direct sum of V3 and a new irreducible representation V4 of dimension 8. Write down the character of V4. From the work you have done so far, show that the 6 irreducible representations of G are of dimension 1,6,7,8, 3 and 3 respectively. Of these, only the characters of the two irreducible three-dimensional representations have not been computed.


12. (optional) Write down the character for the 6 dimensional representation which is the direct sum of the two irreducible three-dimensional representations attached to G.