Lecture notes for Introduction to Representation Theory

Corollary 4.49 easily implies that the coefficient of x +χ y +χ is 1. Indeed, if ε ⇒= 1 is a permutation
in S N , the coefficient of this monomial in Q is obviously
1
zero.
(1−xj y (j) )
Remark. For partitions ∂ and µ of n, let us say that ∂ µ or µ ⇔ ∂ if µ − ∂ is a sum of
vectors of the form e i − e j , i < j (called positive roots). This is a partial order, and µ ⇔ ∂ implies
µ ⊂ ∂. It follows from Theorem 4.47 and its proof that
ν = µ≥ K µν Uµ .
This implies that the Kostka numbers K µ vanish unless µ ⇔ ∂.
4.16 Problems
In the following problems, we do not make a distinction between Young diagrams and partitions.
Problem 4.50. For a Young diagram µ, let A(µ) be the set of Young diagrams obtained by adding
a square to µ, and R(µ) be the set of Young diagrams obtained by removing a square from µ.
(a) Show that Res Sn V µ = R(µ) V .
S n−1
(b) Show that Ind Sn
S n−1
V µ = A(µ) V .
Problem 4.51. The content c(∂) of a Young diagram ∂ is the sum ⎨ ⎨ j
⎨
j i=1
(i − j). Let C =
i