## Lectures in Representation Theory

Last update: 20 August 2013

## Lecture 14

Here, we use the fact that in the symmetric group $\sigma$ is conjugate to ${\sigma }^{-1}\text{.}$

Lemma 2.35 $|{𝒞}_{\mu }|=\frac{|{S}_{m}|}{\mu ?}\text{.}$

 Proof. Let $\mu =\left({1}^{{\mu }_{1}}{2}^{{\mu }_{2}}{3}^{{\mu }_{3}}\cdots \right)$ be a partition of $m\text{.}$ The conjugacy class of ${S}_{m}$ labeled by $\mu$ consists of all permutations with cycle structure $\mu ,$ i.e., permutations having ${\mu }_{1}$ one-cycles, ${\mu }_{2}$ two-cycles, etc. We count $|{𝒞}_{\mu }|$ 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 $k\text{-cycles}$ in this permutation. There are $k$ cyclic permutations of length $k$ of this cycle all of which give the same permutation. Moreover, there are ${m}_{k}!$ ways of reordering the different $k\text{-cycles,}$ so $|𝒞μ|= m! 1m1m2! 2m2m1! 3m3m3!⋯ =m!μ?= |Sm|μ?.$ $\square$

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

We now return to the representation theory of ${S}_{m}\text{.}$ From the definition (2.13) of the weighted trace wtr of ${S}_{n},$ we have $wtr(σ)=∑λ⊢m trλ(σ)mλ(x)$ The trace tr is the trace of some representation of ${S}_{m},$ so it is a character of ${S}_{n}$ and can be written in terms of the irreducible ${S}_{m}\text{-characters}$ ${\chi }^{\nu }$ as $trλ=∑ν⊢m Kν,λχν. (2.37)$ The coefficient ${K}_{\nu ,\lambda }$ is a non-negative integer known as a Kostka coefficient. Thus, we have $pμ(x) = ∑λ⊢mℓ(λ)≤n trλ(μ) mλ(x)from (2.17) = ∑λ⊢mℓ(λ)≤n ( ∑νKν,λ χν(μ) ) ( ∑β Kλ,β-1 sβ(x) ) from (2.37) and (2.31) = ∑β [ ∑ν (∑λKν,λKλ,β-1) χν(μ) ] ⏟ηβ(ν) sβ(x)from (2.31).$ We know that ${\sum }_{\lambda }{K}_{\nu ,\lambda }{K}_{\lambda ,\beta }^{-1}\in ℤ\text{.}$ Denote it by ${c}_{\nu ,\beta }\text{.}$ Then we have $ηβ=∑ν⊢m cν,βχν.$ This holds for all $\beta ⊢m$ with $\ell \left(\beta \right)\le n\text{.}$ We make the assumption that $n\ge m,$ so that the matrix $C=\left({c}_{\nu ,\beta }\right)$ 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 ${\delta }_{\beta ,\gamma }={\sum }_{\alpha }{c}_{\alpha \beta }{c}_{\alpha \gamma },$ or $I=C{C}^{t}\text{.}$ In other words, $C\in O\left(N,ℤ\right)$ where $N$ is the number of partitions of $m\text{.}$

Definition 2.38 The hyper octahedral group $W{B}_{N}$ 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 $-1\text{.}$

Homework Problem 2.39

 (1) $W{B}_{N}=O\left(N,ℤ\right)\text{.}$ (2) $W{B}_{N}$ can be given by generators ${s}_{1},{s}_{2},\dots ,{s}_{N}$ and relations $sisj= sjsi |i-j|>1, sisi+1si= si+1sisi+1 1≤i≤N-2, sN-1sN sN-1sN= sNsN-1 sNsN-1$ (3) Compute the characters of $W{B}_{N}\text{.}$ (Hint: you need to use two alphabets.)

Thus $\left({c}_{\alpha \beta }\right)$ is a signed permutation, and so we have ${\eta }^{\beta }=±{\chi }^{\beta }\text{.}$ Therefore, we have proved Frobenius formula up to sign. That is, we have shown $pμ(x)= ∑β⊢n± χβ(μ) sβ(x).$ To prove that the sign must be positive, we again consider the representation theory of ${S}_{m}\text{.}$ We know that $χβ(1m)= χβ(1)=dim Vβ,$ where ${V}^{\beta }$ is the irreducible ${S}_{m}\text{-module}$ indexed by $\beta \text{.}$ In particular ${\chi }^{\beta }\left({1}^{m}\right)>0,$ so if we can show that ${\eta }^{\beta }\left({1}^{m}\right)>0,$ then the Frobenius formula is proved. From (2.31), we have $p(1m)(x)= ∑β⊢mnβ (1m)sβ(x),$ 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 $\lambda ,\mu$ are partitions, the we say that $\lambda \subseteq \mu$ if ${\lambda }_{i}\le {\mu }_{i}$ for all $i\text{.}$

Example 2.41 If $\lambda =\left(5,3,3,2,1\right)$ and $\mu =\left(6,6,4,3,1\right),$ then $\lambda \subseteq \mu \text{.}$ We picture this as $.$ The whole picture is the partition $\mu ,$ and the boxes that are not dotted make up the partition $\lambda \text{.}$

Definition 2.42 If $\lambda \subseteq \mu ,$ then the skew diagram $\mu /\lambda$ is the set-theoretic difference of the Ferrers diagrams of $\mu$ and $\lambda \text{.}$

In example 2.41, the dotted boxes form $\mu /\lambda \text{.}$ Two boxes in $\mu /\lambda$ 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.