Lectures in Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 20 August 2013

Lecture 14

Here, we use the fact that in the symmetric group σ is conjugate to σ-1.

Lemma 2.35 |𝒞μ|= |Sm|μ?.


Let μ=(1μ12μ23μ3) be a partition of m. The conjugacy class of Sm labeled by μ consists of all permutations with cycle structure μ, i.e., permutations having μ1 one-cycles, μ2 two-cycles, etc. We count |𝒞μ| by first writing down an element of this class. There are m! ways of doing this, since there are m! ways of listing the m numbers. Now consider the k-cycles in this permutation. There are k cyclic permutations of length k of this cycle all of which give the same permutation. Moreover, there are mk! ways of reordering the different k-cycles, so |𝒞μ|= m! 1m1m2! 2m2m1! 3m3m3! =m!μ?= |Sm|μ?.

Thus orthogonality of characters for the symmetric group can be written as δλ,μ= νm χλ(ν) χμ(ν) ν? . (2.36)

We now return to the representation theory of Sm. From the definition (2.13) of the weighted trace wtr of Sn, we have wtr(σ)=λm trλ(σ)mλ(x) The trace tr is the trace of some representation of Sm, so it is a character of Sn and can be written in terms of the irreducible Sm-characters χν as trλ=νm Kν,λχν. (2.37) The coefficient Kν,λ is a non-negative integer known as a Kostka coefficient. Thus, we have pμ(x) = λm(λ)n trλ(μ) mλ(x)from (2.17) = λm(λ)n ( νKν,λ χν(μ) ) ( β Kλ,β-1 sβ(x) ) from (2.37) and (2.31) = β [ ν (λKν,λKλ,β-1) χν(μ) ] ηβ(ν) sβ(x)from (2.31). We know that λKν,λKλ,β-1. Denote it by cν,β. Then we have ηβ=νm cν,βχν. This holds for all βm with (β)n. We make the assumption that nm, so that the matrix C=(cν,β) is square. From (2.32) and and (1.58), we have δρ,τ = μ ηβ(μ) ηγ(μ) μ? = μ (αcαβχα(μ)) (κcκγχκ(μ)μ?) = α,κ cαβ cκγ ( μχα(μ) χκ(μ)μ? ) = α,κcαβ cκγδακ. Therefore, we have δβ,γ=αcαβcαγ, or I=CCt. In other words, CO(N,) where N is the number of partitions of m.

Definition 2.38 The hyper octahedral group WBN is the group of N by N matrices such that

(1) there is exactly one nonzero entry in each row and each column,
(2) this nonzero entry is either 1 or -1.

Homework Problem 2.39

(1) WBN=O(N,).
(2) WBN can be given by generators s1,s2,,sN and relations sisj= sjsi |i-j|>1, sisi+1si= si+1sisi+1 1iN-2, sN-1sN sN-1sN= sNsN-1 sNsN-1
(3) Compute the characters of WBN. (Hint: you need to use two alphabets.)

Thus (cαβ) is a signed permutation, and so we have ηβ=±χβ. Therefore, we have proved Frobenius formula up to sign. That is, we have shown pμ(x)= βn± χβ(μ) sβ(x). To prove that the sign must be positive, we again consider the representation theory of Sm. We know that χβ(1m)= χβ(1)=dim Vβ, where Vβ is the irreducible Sm-module indexed by β. In particular χβ(1m)>0, so if we can show that ηβ(1m)>0, then the Frobenius formula is proved. From (2.31), we have p(1m)(x)= βmnβ (1m)sβ(x), so we are interested in a formula for multiplying power symmetric functions and Schur functions. With this in mind, we make some definitions.

Definition 2.40 If λ,μ are partitions, the we say that λμ if λiμi for all i.

Example 2.41 If λ=(5,3,3,2,1) and μ=(6,6,4,3,1), then λμ. We picture this as . The whole picture is the partition μ, and the boxes that are not dotted make up the partition λ.

Definition 2.42 If λμ, then the skew diagram μ/λ is the set-theoretic difference of the Ferrers diagrams of μ and λ.

In example 2.41, the dotted boxes form μ/λ. Two boxes in μ/λ are connected if they share a common edge.

Definition 2.43 A border strip is a skew diagram that

(1) contains no two-by-two blocks of boxes, and
(2) is connected; that is, consists of a single sequence of connected boxes.
A border strip has length r if it contains r boxes.

Example 2.44 The dotted boxes in the following diagram form a border strip of length 8. The skew diagram in Example 2.41 does not form a border strip, because it is not connected. The dotted boxes in do not form a border strip, because they contain a two-by-two block of boxes.

Notes and References

This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.

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