18.315  Problem Set 1.  (due Thursday, September 21)

1.  Prove (using only the definition of representations) that the symmetric 
group S_n, n>=2, has exactly two 1-dimensional representations: 
the trivial representation and the sign representation.

2.  Prove that the number of paths of length 2n in the Young graph that
start and end at the 0-th level (and can have up and down edges in any 
order) equals (2n-1)!! := 1*3*...*(2n-1).

The following fact might be useful.  The number of complete matchings
of the set {1,2,...,2n} (i.e., ways to subdivide this set into n pairs) 
equals (2n-1)!!.   (But if you use this fact you need to prove it.)

3. Let X and Y be two matrices that satisfy the relation YX-XY = X.
Prove that Y^m X^n = \sum_{k=0}^m {m \choose k} n^{m-k} X^n Y^k. 

4. Construct explicitly a 2-dimensional irreducible representation 
of S_4 by presenting 2x2 matrices for the 3 adjacent permutations.

5*.   Let F be the directed graph on the set of all sequences of 1's and 2's 
(including the empty sequence o) with edges of the following form  
      (a_1,...,a_k,2^r) --> (a_1,...,a_k,1,2^r) and 
      (a_1,...,a_k,1,2^r) --> (a_1,...,a_k,2^{r+1}) 
for any r,k >=0, where a_1,...,a_k is any sequence of 1's and 2's and 
2^r denotes the sequence with r 2's.  For a vertex v in F, let p(v) be 
the number of directed paths from o to v.
Prove that \sum_v p(v)^2 = n!, where the sum is over all vertices
v = (a_1,...,a_k) in F such that a_1+...+a_k = n.