MA 591 Homework 4 – due 10/13
*Starred problems are optional supplementary exercises.
Homework:
1. A plane partition π is a tableau of positive integers (of partition shape) such that
each row and column is weakly decreasing. Let R(a, b, c) denote the number of plane
partitions that fit in an a × b rectangle and whose entries are all at most c.
(a) Show that R(a, b, c) is symmetric in a, b, and c.
(b) Show that
R(a, b, c) = a+b
a a+b
a−1
a+b
a+1
a+b
a
a−c+1
a−c+2
..
. a+b
..
. a+b
···
···
..
.
···
a+b a+c−1
a+b a+c−2 .. .
. a+b a
(c) *A lozenge is a rhombus formed by joining two equilateral triangles of side length
1. Show that R(a, b, c) is the number of ways to tile an equiangular hexagon with
side lengths a, b, c, a, b, c with lozenges.
Q Q Q
(d) *Show that R(a, b, c) = ai=1 bj=1 ck=1 i+j+k−1
.
i+j+k−2
2. Let δ = (n − 1, n − 2, . . . , 1).
(a) Show that sδ (x1 , . . . , xn ) =
Q
1≤i<j≤n (xi
+ xj ).
(b) Show that sδ/(2) = sδ/(1,1) .
P
3. Show that R sR = sn1 , where R ranges over all 2n−1 ribbons with n boxes (up to
translation).
4. (a) Suppose λ does not contain a hook of size q. Show that λ does not contain any
hook whose size is a multiple of q. (Such a partition is called a q-core.)
(b) Suppose λ is a q-core. Show that sλ lies in the subalgebra of Λ generated by the
power sums pi for i not divisible by q.
⊥
5. *Let s⊥
1 be the linear operator on Λ adjoint to multiplication by s1 , i.e., s1 sλ = sλ/(1) .
∂f
⊥
Show that s1 f = ∂p1 when f ∈ λ = C[p1 , p2 , . . . ] is written as a polynomial in the
power sums.
1