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 10

Proposition 2.6 If ${𝒮}_m$ is defined in terms of diagrams, with $$s_i\leftrightarrow \begin{array}{c} ⋯ i i+1 ⋯ \end{array}$$ then $\{s_1,\dots ,s_{m-1}\}$ generate ${𝒮}_m\text{.}$

		Illustration:
	We will examine the permutation $\sigma =\left(1\hspace{0.17em}4\right)\left(3\hspace{0.17em}6\hspace{0.17em}5\right)\in {𝒮}_6\text{.}$ $$\begin{array}{ccccc}\sigma & =& & =& s3 s4 s5 s1 s2 s3 s1 \end{array}$$ Thus $\sigma =s_3{s}_4{s}_5{s}_1{s}_2{s}_3{s}_1\text{.}$ $\text{“}\square \text{”}$

Definition 2.7 A reduced word for $\sigma \in {𝒮}_m$ is an expression $\sigma =s_{i_1}\cdots s_{i_p}$ of minimal length. The number $p$ is the length of $\sigma ,$ often denoted $\ell \left(\sigma \right)\text{.}$

Recall (or note) that the length of a permutation $\sigma$ is equal to the number of crossings in the diagram for $\sigma$ (with suitable conventions on the diagram – no superfluous crossings, e.g.). The length is also equal to the number of inversions of the permutation. We will also need in what follows the fact that every presentation of a permutation of a product of $s_i$ has its number of generators in the presentation congruent to $\ell \left(\sigma \right)$ modulo two.

Definitions 2.8 The set of inversions of $\sigma \in {𝒮}_m$ is the set $$\text{inv}\left(\sigma \right)=\left\{(i,j)\hspace{0.17em}\right|\hspace{0.17em}1\le i<j\le m,\sigma \left(i\right)>\sigma \left(j\right)\}$$ The sign is $\epsilon \left(\sigma \right)={(-1)}^{\ell \left(\sigma \right)}\text{.}$

Note that in ${𝒮}_m$ each permutation has the same cycle type as its inverse, and therefore for any character $\chi$ one has ${\chi }^{\left({\sigma }^{-1}\right)}=\chi \left(\sigma \right)=\chi \left({\gamma }_{\tau \left(\sigma \right)}\right)\text{.}$ By an abuse of notation, we will sometimes denote $\chi \left({\gamma }_{\mu }\right)$ by $\chi \left(\mu \right)\text{.}$

For the trivial representation of ${𝒮}_m,$ we have $V^{\left(m\right)}:ℂ{𝒮}_m\to M_1\left(ℂ\right):\sigma \mapsto \left(1\right)\text{.}$ The associated character is ${\chi }^{\left(m\right)}:ℂ{𝒮}_m\to ℂ:\sigma \mapsto 1\text{.}$ One has that ${\chi }^{\left(m\right)}\left({\gamma }_{\mu }\right)=1\text{.}$

Now, let us consider the alternating representation of ${𝒮}_m\text{.}$ Denoted by $V^{\left(1^m\right)},$ it is generated by sending each $s_i$ to the matrix $(-1)\text{.}$ One has $$\begin{array}{c}\begin{array}{cccc}{V}^{\left(1^m\right)}:& ℂ{𝒮}_m& ⟼& M_1\left(ℂ\right)\\ & s_i& ⟼& (-1)\end{array}\\ \\ \sigma =s_{i_1}\cdots s_{i_{\ell \left(\sigma \right)}}\mapsto {(-1)}^{\ell \left(\sigma \right)}\end{array}$$ (Again, note that $V^{\left(1^m\right)}\left(\sigma \right)$ is independent of the choice of presentation of $\sigma \text{.)}$ This representation gives rise to the character $$\begin{array}{cccc}{\chi }^{\left(1^m\right)}:& ℂ{𝒮}_m& ⟶& ℂ\\ & s_i& ⟼& -1\\ & \sigma & ⟼& {(-1)}^{\ell \left(\sigma \right)}\end{array}$$ Since characters are class functions, we only need consider their behavior on cycle structures; we have that ${\chi }^{\left(1^m\right)}\left({\gamma }_{\mu }\right)={(-1)}^{\ell \left({\gamma }_{\mu }\right)}={(-1)}^{m-\ell \left(\mu \right)},$ since $$\begin{array}{ccc}\ell \left({\gamma }_{\mu }\right)& =& \ell ({\gamma }_{{\mu }_1}\times {\gamma }_{{\mu }_2}\times \cdots \times {\gamma }_{{\mu }_k})\\ & =& ({\mu }_1-1)+({\mu }_2-1)\cdots ({\mu }_k-1)\\ & =& m-\ell \left(\mu \right)\end{array}$$

Recall that the minimal central idempotent for the trivial representation was $$z_{\left(m\right)}=\frac{1}{m!}\sum _{\sigma \in {𝒮}_m}\sigma$$ For the alternating representation, we have $$\begin{array}{ccc}{z}_{\left(1^m\right)}& =& \frac{1}{m!}\sum _{\sigma \in {𝒮}_m}{d}_{\left(1^m\right)}{\chi }^{\left(1^m\right)}\left({\sigma }^{-1}\right)\sigma \\ & =& \frac{1}{m!}\sum _{\sigma \in {𝒮}_m}{(-1)}^{\ell \left(\sigma \right)}\sigma \\ & =& \frac{1}{m!}\sum _{\sigma \in {𝒮}_m}\epsilon \left(\sigma \right)\sigma \end{array}$$

Homework Problem 2.9 Find analogs of the above representations, characters, and idempotents for the Hecke algebras.

We have seen the two one-dimensional representations of ${𝒮}_m\text{.}$ Now we will consider the permutation representation, wherein $V:ℂ{𝒮}_m\to M_m\left(ℂ\right)$ sends each permutation to the appropriate permutation matrix. This representation (unlike the trivial and alternating representations) is not irreducible; in fact $V\simeq V^{\left(m\right)}\oplus V^{(m-1,1)}\text{.}$ Note that the determinant of the permutation matrix $\sigma$ is ${(-1)}^{\ell \left(\sigma \right)}=\epsilon \left(\sigma \right),$ so by taking the composition of the permutation representation and the determinant map one (essentially) obtains the alternating representation.

Now we consider the representation of ${𝒮}_m$ on words of length $m\text{.}$ Let $v_1,\dots ,v_n$ be $n$ letters (i.e. indeterminate symbols), and let $$V^{\otimes m}=ℂ\text{-span of}\hspace{0.17em}\left\{v_{i_1}\cdots v_{i_m}\hspace{0.17em}\right|\hspace{0.17em}1\le i_1,\dots ,i_m\le n\}$$ $ℂ{𝒮}_m$ acts on $V^{\otimes m}$ from the right by $$v_{i_1}\cdots v_{i_m}·\sigma =v_{i_{\sigma \left(1\right)}}\cdots v_{i_{\sigma \left(m\right)}}$$

Definition 2.10 Let $x_1,\dots ,x_n$ be commuting variables. Define the weight of $v_{i_1}\cdots v_{i_m}$ to be $\text{wt}\left(v_{i_1}\cdots v_{i_m}\right)=x_{i_1}\cdots x_{i_m}\text{.}$

Note, then, that upon re-ordering, $\text{wt}\left(v_{i_1}\cdots v_{i_m}\right)=x_1^{{\lambda }_1}\cdots x_n^{{\lambda }_n},$ where ${\lambda }_j$ is the number of occurrences of the letter $v_j$ in the word $v_{i_1}\cdots v_{i_m}\text{.}$ Note also that $\text{wt}\left(v_{i_1}\cdots v_{i_m}\sigma \right)=\text{wt}\left(v_{i_1}\cdots v_{i_m}\right)$ for $\sigma \in {𝒮}_m\text{.}$

Definition 2.11 A sequence $\lambda =({\lambda }_1,\dots ,{\lambda }_n)$ such that ${\lambda }_j\ge 0$ and $\sum {\lambda }_j=m$ is called a composition of $m,$ denoted $\lambda \models m\text{.}$

Definition 2.12 Fix a composition $\lambda \models m\text{.}$ The following are all submodules of $V^{\otimes m}\text{.}$ $$1{\uparrow }_{s\ell }^{{𝒮}_m}=ℂ\text{-span of}\hspace{0.17em}\left\{v_{i_1}\cdots v_{i_m}\hspace{0.17em}\right|\hspace{0.17em}\text{wt}\left(v_{i_1}\cdots v_{i_m}\right)=x^{\lambda }\}$$

The fact that the above are submodules follows from the stability of the weight operator under the action of $ℂ{𝒮}_m\text{.}$

Definition 2.13 The weighted trace of a permutation $\sigma \in {𝒮}_m,$ denoted $\text{wtr}\left(\sigma \right),$ is $$\text{wtr}\left(\sigma \right)=\sum _{v_{i_1}\cdots v_{i_m}}\left(v_{i_1}\cdots v_{i_m}\sigma {|}_{v_{i_1}\cdots v_{i_m}}\right)x_{i_1}\cdots x_{i_m}$$

We would like to compute the characters of the representations $1{\uparrow }_{s\ell }^{{𝒮}_m}\text{.}$ Note that $$\begin{array}{ccc}{\text{tr}}_{\lambda }\left(\sigma \right)& =& \sum _{\underset{\text{wt}\left(v_{i_1}\cdots v_{i_m}\right)=x^{\lambda }}{v_{i_1}\cdots v_{i_m}}}{v}_{i_1}\cdots v_{i_m}\sigma {|}_{v_{i_1}\cdots v_{i_m}}\\ & =& \text{wtr}\left(\sigma \right){|}_{x^{\lambda }}\end{array}$$ Now, since the trace tr of a permutation is completely determined by the trace of the associated cycle structure, the same is true of the weighted trace wtr. Thus to compute the traces ${\text{tr}}_{\lambda },$ it suffices to compute the weighted traces $\text{wtr}\left({\gamma }_{\mu }\right)\text{.}$

Lemma 2.14 If $\mu =({\mu }_1,\dots ,{\mu }_k)⊢m,$ so ${\gamma }_{\mu }={\gamma }_{{\mu }_1}\times \cdots \times {\gamma }_{{\mu }_k},$ then $\text{wtr}\left({\gamma }_{\mu }\right)=\text{wtr}\left({\gamma }_{{\mu }_1}\right)\text{wtr}\left({\gamma }_{{\mu }_2}\right)\cdots \text{wtr}\left({\gamma }_{{\mu }_k}\right)\text{.}$

		Proof.
	$$\begin{array}{ccc}\text{wtr}\left({\gamma }_{\mu }\right)& =& \sum _{1\le i_1,\dots i_m\le m}\left(v_{i_1}\cdots v_{i_m}\right)({\gamma }_{{\mu }_1}\times {\gamma }_{{\mu }_2}\times \cdots \times {\gamma }_{{\mu }_k}){|}_{v_{i_1}\cdots v_{i_m}}{x}_{i_1}\cdots x_{i_m}\\ & =& \left(\sum _{i_1,\dots ,i_{{\mu }_1}}\left(v_{i_1}\cdots v_{i_{{\mu }_1}}\right){\gamma }_{{\mu }_1}{|}_{v_{i_1}\cdots v_{i_{{\mu }_1}}}{x}_{i_1}\cdots x_{i_{{\mu }_1}}\right)\times \\ & & \times (\sum _{i_{{\mu }_1+1}},\dots ,i_{{\mu }_2}\left(v_{i_{{\mu }_1+1}}\cdots v_{i_{{\mu }_2}}\right){\gamma }_{{\mu }_2}{|}_{v_{i_{{\mu }_1+1}}\cdots v_{i_{{\mu }_2}}}{x}_{i_{{\mu }_1+1}}\cdots x_{i_{{\mu }_2}})\cdots \\ & =& \text{wtr}\left({\gamma }_{{\mu }_1}\right)\text{wtr}\left({\gamma }_{{\mu }_2}\right)\cdots \end{array}$$ The above works since ${\gamma }_{{\mu }_1}$ only acts on the first ${\mu }_1$ many characters, ${\gamma }_{{\mu }_2}$ only acts on the ${\mu }_1+1$ through ${\mu }_2$ characters, etc.. $\square$

Proposition 2.15 If $r>0,$ and $r\in \mathbb{P},$ then $\text{wtr}\left({\gamma }_r\right)=x_1^r+\cdots +x_n^r\text{.}$

		Proof.
	Recall that $v_{i_1}{v}_{i_2}\cdots v_{i_r}{\gamma }_r=v_{i_r}{v}_{i_1}\cdots v_{i_{r-1}}\text{.}$ Thus $$\begin{array}{ccc}\text{wtr}\left({\gamma }_r\right)& =& \sum _{1\le i_1,\dots ,i_r\le n}{v}_{i_1}\cdots v_{i_r}·{\gamma }_r{|}_{v_{i_1}\cdots v_{i_r}}{x}_{i_1}\cdots x_{i_r}\\ & =& \sum _{1\le i_1,\dots ,i_r\le n}{v}_{i_r}{v}_{i_1}\cdots v_{i_{r-1}}{|}_{v_{i_1}\cdots v_{i_r}}{x}_{i_1}\cdots x_{i_r}\\ & =& \sum _{i_r=i_1=i_2=\cdots =i_{r-1}}{x}_{i_1}\cdots x_{i_r}\\ & =& \sum _{1\le i_1\le n}{x}_{i_1}^r\\ & =& x_1^r+x_2^r+\cdots +x_n^r\text{.}\end{array}$$ $\square$

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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