Problem set 4
Due date: 04/22/2016
Before you start working on the homework problems read the extended
syllabus1 carefully and check which problems you need to solve.
1. Practice problems
P1. Show that
U n Dn = (U D − (n − 1)I)(U D − (n − 2)I) . . . U D,
where U and D are the linear transformations associated to the Young-lattice,
where both sides operate on the vectorspace Rj for som j.
P2. Let f ∈ Fp [x1 , . . . , xn ] be a multivariate polynomial of degree n(p − 1).
Let
∫
∑
f=
f (a).
Show that
∫
a∈Fn
p
f depends only on the term where every exponent is p − 1.
P3. Let F be an arbitrary field and let A be an n × n matrix over F. Assume
that the permanent of A is non-zero. Show that for arbitrary vector b ∈ Fn
and sets S1 , . . . , Sn ⊆ F for which |Si | = 2 for all i, there exists a vector
x = (x1 , . . . , xn ) ∈ S1 × S2 × · · · × Sn such that all coordinates of Ax and b are
different.
2. Homework problems
1. Let π ∈ Sn be a random permutation uniformly chosen from Sn , and let
us assume that the RSK-algorithm maps π to the standard Young-tableauxs
(P, Q). What’s the probability that the elements 1, 2, . . . , k all appear in the
first row of π? Let f (π) be the largest k such that 1, 2, . . . , k all appear in the
first row of π. What’s the probability that f (π) = r? Let E(n) denote the
expected value of f (π), i. e.,
1 ∑
E(n) =
f (π).
n! π∈S
n
Find limn→∞ E(n)?
1
The extended syllabus can be found at
http://math.mit.edu/~csikvari/extended_syllabus_algebraic_combinatorics.pdf
1
2
2. Let c(λ) be the number of distinct parts of the partition λ. (For instance,
c((4, 4, 2, 2, 1)) = 3 as 4, 2, 1 are the distinct parts.) Show that
∑
c(λ) = p(0) + p(1) + · · · + p(n − 1),
λ⊢n
where p(k) is the number of partitions of k.
3. Let w be a balanced word in U and D, i. e., the same number of U ’s and
D’s. For instance, U U DU DDU D is a balanced word. Regard w as a linear
transformation on the
in the usual way. Then w maps RYn
∑Young-lattice
to itself. Show that
f λ eλ is an eigenvector of w. Find the corresponding
eigenvalue.
4. The planes H1 , . . . , Hk cover all lattice points of the cube [0, 1, . . . , n]3 except
the point (0, 0, 0). Show that k ≥ 3n.
5. Show that
(
⌊n/2⌋
∑
f
(n−i,i)
)
n
,
⌊n/2⌋
=
i=0
where f
(n−i,i)
is the number of standard Young-tableauxs with shape (n − i, i).
6. Show that any two balanced words (see exercise 3) commute.
7. Show that
a
∑
(
k
(−1)
k=−a
a+b
a+k
)(
b+c
b+k
)(
c+a
c+k
)
(
)
a+b+c
=
.
a, b, c
8. Given a graph G = (V, E) and at every vertex given a set D(v) ⊆
{0, 1, . . . , d(v)} such that |D(v)| ≥ d(v)/2+1, where d(v) is the degree of vertex
v. Show that there exists a spanning subgraph H of G such that dH (v) ∈ D(v)
for every vertex v. (A spanning subgraph H contains all vertices of the graph
G and some of the edges.)
{ }
9. Let nk be the number of ways one can decompose the set {1, 2, . . . , n}
into k disjoint non-empty sets. Show that the polynomial
n { }
∑
n k
x
k
k=1
3
has only real zeros. Deduce that
{ n }2 { n } { n }
≥
k
k−1
k+1
for 1 ≤ k ≤ n.
10. Let p be a prime. A set A intersects all lines of the affine plane F2p .
Show that |A| ≥ 2p − 1. (The lines of the affine plane F2p are the following:
ℓa,b = {(x, y) | y = ax + b, x ∈ Fp } and ℓa = {(a, x) | x ∈ Fp }, where a, b ∈ Fp ,
so the affine plane F2p has p2 + p lines.)
11. Let k = p−1
. Show that for arbitrary d1 , . . . , dk ∈ F∗p there exists a
2
permutation a1 , b1 , a2 , b2 , . . . , ak , bk of F∗p such that ai − bi = di .
12. What is the constant term of the product
(
)(
)
)
∏ (
xi
xj
1
1−
(1 − xi xj ) 1 −
?
1−
xj
xi
xi xj
1≤i<j≤n