µi− i if µ is obtained by adding a box to the i-th row of λ

L165100 - Fall 2010 - Homework 2

1. Prove that, for any Young diagramλ of n boxes, we have X

λlµ

f^µ= (n + 1)f^λ,

where the sum is over all Young diagramsµ obtained by adding a box to λ.

2. Prove that, for any Young diagramλ, we have X

λlµ

c(λ, µ)f^µ= 0,

where c(λ, µ) = µi− i if µ is obtained by adding a box to the i-th row of λ. (Hint: show that the linear map V on KY given by

V(λ) =X

λlµ

c(λ, µ)µ

commutes with the usual lowering operator D on KY . )

3. Let P be a finite poset whose Hasse diagram is a rooted tree, that is, P has a unique minimal elements ˆ0, and, for every x ∈ P, the interval [ˆ0, x] is a chain. Show that the number N of linear extensions of P (that is, the number of total orderings of P compatible with the given partial order) is given by the following hook-length type formula: N= |P|!/Πx∈Ph(x), where h(x) is the number of elements y ∈ P such that x ≤ y.

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