Lectures in Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $\left|{𝒞}_{\mu }\right|={\displaystyle \frac{\left|S_m\right|}{\mu ?}\text{.}}$

		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 $\left|{𝒞}_{\mu }\right|$ 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 $$\left|{𝒞}_{\mu }\right|=\frac{m!}{1^{m_1}{m}_2!2^{m_2}{m}_1!3^{m_3}{m}_3!\cdots }=\frac{m!}{\mu ?}=\frac{\left|S_m\right|}{\mu ?}\text{.}$$ $\square$

Thus orthogonality of characters for the symmetric group can be written as $$\begin{array}{cc}{\delta }_{\lambda ,\mu }=\sum _{\nu ⊢m}\frac{{\chi }^{\lambda }\left(\nu \right){\chi }^{\mu }\left(\nu \right)}{\nu ?}\text{.}& \text{(2.36)}\end{array}$$

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 $$\text{wtr}\left(\sigma \right)=\sum _{\lambda ⊢m}{\text{tr}}_{\lambda }\left(\sigma \right)m_{\lambda }\left(x\right)$$ 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 $$\begin{array}{cc}{\text{tr}}_{\lambda }=\sum _{\nu ⊢m}{K}_{\nu ,\lambda }{\chi }^{\nu }\text{.}& \text{(2.37)}\end{array}$$ The coefficient $K_{\nu ,\lambda }$ is a non-negative integer known as a Kostka coefficient. Thus, we have $$\begin{array}{ccc}{p}_{\mu }\left(x\right)& =& \sum _{\underset{\ell \left(\lambda \right)\le n}{\lambda ⊢m}}{\text{tr}}_{\lambda }\left(\mu \right)m_{\lambda }\left(x\right)\text{from (2.17)}\\ & =& \sum _{\underset{\ell \left(\lambda \right)\le n}{\lambda ⊢m}}\left(\sum _{\nu }{K}_{\nu ,\lambda }{\chi }^{\nu }\left(\mu \right)\right)\left(\sum _{\beta }{K}_{\lambda ,\beta }^{-1}{s}_{\beta }\left(x\right)\right)\text{from (2.37) and (2.31)}\\ & =& \sum _{\beta }\underset{\underset{{\eta }^{\beta }\left(\nu \right)}{⏟}}{\left[\sum _{\nu }\left(\sum _{\lambda }{K}_{\nu ,\lambda }{K}_{\lambda ,\beta }^{-1}\right){\chi }^{\nu }\left(\mu \right)\right]}{s}_{\beta }\left(x\right)\text{from (2.31).}\end{array}$$ We know that ${\sum }_{\lambda }{K}_{\nu ,\lambda }{K}_{\lambda ,\beta }^{-1}\in \mathbb{Z}\text{.}$ Denote it by $c_{\nu ,\beta }\text{.}$ Then we have $${\eta }^{\beta }=\sum _{\nu ⊢m}{c}_{\nu ,\beta }{\chi }^{\nu }\text{.}$$ 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 $$\begin{array}{ccc}{\delta }_{\rho ,\tau }& =& \sum _{\mu }\frac{{\eta }^{\beta }\left(\mu \right){\eta }^{\gamma }\left(\mu \right)}{\mu ?}\\ & =& \sum _{\mu }\left(\sum _{\alpha }{c}_{\alpha \beta }{\chi }^{\alpha }\left(\mu \right)\right)\left(\sum _{\kappa }\frac{c_{\kappa \gamma }{\chi }^{\kappa }\left(\mu \right)}{\mu ?}\right)\\ & =& \sum _{\alpha ,\kappa }{c}_{\alpha \beta }{c}_{\kappa \gamma }\left(\sum _{\mu }{\chi }^{\alpha }\left(\mu \right)\frac{{\chi }^{\kappa }\left(\mu \right)}{\mu ?}\right)\\ & =& \sum _{\alpha ,\kappa }{c}_{\alpha \beta }{c}_{\kappa \gamma }{\delta }_{\alpha \kappa }\text{.}\end{array}$$ Therefore, we have ${\delta }_{\beta ,\gamma }={\sum }_{\alpha }{c}_{\alpha \beta }{c}_{\alpha \gamma },$ or $I=C{C}^t\text{.}$ In other words, $C\in O(N,\mathbb{Z})$ 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)	$W{B}_N=O(N,\mathbb{Z})\text{.}$
(2)	$W{B}_N$ can be given by generators $s_1,s_2,\dots ,s_N$ and relations $$\begin{array}{cc}{s}_i{s}_j=s_j{s}_i& |i-j|>1,\\ s_i{s}_{i+1}{s}_i=s_{i+1}{s}_i{s}_{i+1}& 1\le i\le N-2,\\ s_{N-1}{s}_N{s}_{N-1}{s}_N=s_N{s}_{N-1}{s}_N{s}_{N-1}\end{array}$$
(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 ${\eta }^{\beta }=\pm {\chi }^{\beta }\text{.}$ Therefore, we have proved Frobenius formula up to sign. That is, we have shown $$p_{\mu }\left(x\right)=\sum _{\beta ⊢n}\pm {\chi }^{\beta }\left(\mu \right)s_{\beta }\left(x\right)\text{.}$$ To prove that the sign must be positive, we again consider the representation theory of $S_m\text{.}$ We know that $${\chi }^{\beta }\left(1^m\right)={\chi }^{\beta }\left(1\right)=\text{dim}\hspace{0.17em}{V}^{\beta },$$ 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 $$p_{\left(1^m\right)}\left(x\right)=\sum _{\beta ⊢m}{n}^{\beta }\left(1^m\right)s_{\beta }\left(x\right),$$ 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 $\lambda =(5,3,3,2,1)$ and $\mu =(6,6,4,3,1),$ then $\lambda \subseteq \mu \text{.}$ We picture this as $$\begin{array}{c} \end{array}\text{.}$$ 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.

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