Mathematics 501
Homework (due Nov 15)
A. Hulpke
53) Let Nλµ be the set of coefficients expressing the completely symmetric polynomials
hλ in terms of the basic polynomials:
hλ =
∑ Nλµ mµ
µ|=n
a) Let p be the number of partitions of n. Show that Nλµ is equal to the number of matrices
p× p
A ∈ Z≥0 which have row sums λ and column sums µ. (E.g. if λ = (2, 1) and µ = (1, 1, 1),
there are three such matrice, namely:

 
 

0 1 1
1 1 0
1 0 1
 0 0 1 , 0 1 0 , 1 0 0 ;
0 0 0
0 0 0
0 0 0
thus N(2,1),(1,1,1) = 3.
b) Show that Nλµ = Nµλ .
c) We define a bilinear form on Span(mλ ) by setting
(mλ , hµ ) = δλµ
and extending by linearity. Show that (hλ , hµ ) = (hµ , hλ ) and conclude that the form is a
scalar product.
(One can show that with respect to this scalar product the power sums pλ are an orthogonal basis and the Schur polynomials sλ are an orthonormal basis.
54) Let F be a field (e.g. R = Q) and G = GLn (F) the set of all invertible n × n matrices
over F.
a) Show that G acts on the subspaces of (the row space) Fn .
b) What are the orbits under this action?
c) Let W = Span(b1 , b2 , b3 ) the subspace spanned by the first 3 vectors in the standard
basis of Fn . Describe StabG (W).
d) Assuming F is finite, can you describe generators for StabG (W).
55) Let G be a group acting transitively (i.e. all points are in the same orbit) on the set
Ω. We define an action of G on Ω × Ω by setting (δ, ω )g := (δ g , ω g ). Show that there is a
bijection between the orbits of G on Ω × Ω and the orbits of StabG (ω ) on Ω.
56) For subgroups H , K ≤ G and g ∈ G we denote the double coset of H and K with
representative g by
HgK = {hgk | h ∈ H , k ∈ K}
a) Show that the double cosets for H and K give a partition of G.
b) Show that HgK is a union of right cosets of H.
c) Suppose that G acts on Ω and that H = StabG (ω ) for some ω ∈ Ω. Show that the double
cosets HgK correspond to the orbits of K on ω G .
57) How many actions up to equivalence does S4 have on a set of 7 points? You may
use without proof that S4 has only one subgroup of order 12 (namely A4 ), one class of
subgroups of order 8 (namely D8 ) and three classes of subgroups of order 4 (namely C4 =
h(1, 2, 3, 4)i, 2 × 2 = h(1, 2), (3, 4)i and V4 = h(1, 2)(3, 4), (1, 3)(2, 4)i).
58) Describe the symmetries of the following graph. Determine a set of generators for
its automorphism group:
4a2
3b3
4a1
3b1
3c
4b1
3a
4b
4b2
7
5d
6d
6b
5b2
6a1
6a2
7b
6a
6b1
2a1
7c
6
5b
5c
2a
7d
5
5a2
5b1
2a2
2
1
5a
2b1
2b
4
4d
2b2
2c
2d
3
4c
5a1
3a1
3b
3d
4a
3a3
7b2
7a
7b1
6c
6b2
7a1
7a2