## Lectures in Representation Theory

Last update: 20 August 2013

## Lecture 10

Proposition 2.6 If ${𝒮}_{m}$ is defined in terms of diagrams, with $si↔ ⋯ i i+1 ⋯$ then $\left\{{s}_{1},\dots ,{s}_{m-1}\right\}$ generate ${𝒮}_{m}\text{.}$

 Illustration: We will examine the permutation $\sigma =\left(1 4\right)\left(3 6 5\right)\in {𝒮}_{6}\text{.}$ $σ = = s3 s4 s5 s1 s2 s3 s1$ 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 $inv(σ)= { (i,j) | 1≤i σ(j) }$ The sign is $\epsilon \left(\sigma \right)={\left(-1\right)}^{\ell \left(\sigma \right)}\text{.}$

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 ${V}^{\left(m\right)}:ℂ{𝒮}_{m}\to {M}_{1}\left(ℂ\right):\sigma ↦\left(1\right)\text{.}$ The associated character is ${\chi }^{\left(m\right)}:ℂ{𝒮}_{m}\to ℂ:\sigma ↦1\text{.}$ 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 $\left(-1\right)\text{.}$ One has $V(1m): ℂ𝒮m ⟼ M1(ℂ) si ⟼ (-1) σ=si1⋯ siℓ(σ) ↦(-1)ℓ(σ)$ (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 $χ(1m): ℂ𝒮m ⟶ ℂ si ⟼ -1 σ ⟼ (-1)ℓ(σ)$ Since characters are class functions, we only need consider their behavior on cycle structures; we have that ${\chi }^{\left({1}^{m}\right)}\left({\gamma }_{\mu }\right)={\left(-1\right)}^{\ell \left({\gamma }_{\mu }\right)}={\left(-1\right)}^{m-\ell \left(\mu \right)},$ since $ℓ(γμ) = ℓ ( γμ1× γμ2×⋯× γμk ) = (μ1-1)+ (μ2-1)⋯ (μk-1) = m-ℓ(μ)$

Recall that the minimal central idempotent for the trivial representation was $z(m)=1m! ∑σ∈𝒮mσ$ For the alternating representation, we have $z(1m) = 1m!∑σ∈𝒮m d(1m)χ(1m) (σ-1)σ = 1m!∑σ∈𝒮m (-1)ℓ(σ)σ = 1m!∑σ∈𝒮m ε(σ)σ$

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 $V\simeq {V}^{\left(m\right)}\oplus {V}^{\left(m-1,1\right)}\text{.}$ Note that the determinant of the permutation matrix $\sigma$ is ${\left(-1\right)}^{\ell \left(\sigma \right)}=\epsilon \left(\sigma \right),$ 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 $V⊗m=ℂ-span of { vi1⋯vim | 1≤i1,… ,im≤n }$ $ℂ{𝒮}_{m}$ acts on ${V}^{\otimes m}$ from the right by $vi1⋯vim· σ=viσ(1) ⋯viσ(m)$

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 $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{n}\right)$ 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 ${V}^{\otimes m}\text{.}$ $1↑sℓ𝒮m=ℂ -span of { vi1⋯ vim | wt(vi1⋯vim) =xλ }$

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 $wtr(σ)= ∑vi1⋯vim (vi1⋯vimσ|vi1⋯vim) xi1⋯xim$

We would like to compute the characters of the representations $1{↑}_{s\ell }^{{𝒮}_{m}}\text{.}$ Note that $trλ(σ) = ∑vi1⋯vimwt(vi1⋯vim)=xλ vi1⋯vimσ |vi1⋯vim = wtr(σ)|xλ$ 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 $\mu =\left({\mu }_{1},\dots ,{\mu }_{k}\right)⊢m,$ so ${\gamma }_{\mu }={\gamma }_{{\mu }_{1}}×\cdots ×{\gamma }_{{\mu }_{k}},$ 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. $wtr(γμ) = ∑1≤i1,…im≤m (vi1⋯vim) (γμ1×γμ2×⋯×γμk) |vi1⋯vim xi1⋯xim = ( ∑i1,…,iμ1 (vi1⋯viμ1) γμ1 |vi1⋯viμ1 xi1⋯xiμ1 ) × × ( ∑iμ1+1 ,…,iμ2 (viμ1+1⋯viμ2) γμ2 |viμ1+1⋯viμ2 xiμ1+1⋯ xiμ2 ) ⋯ = wtr(γμ1) wtr(γμ2)⋯$ 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 $r\in ℙ,$ 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 $wtr(γr) = ∑1≤i1,…,ir≤n vi1⋯vir·γr |vi1⋯vir xi1⋯xir = ∑1≤i1,…,ir≤n virvi1⋯vir-1 |vi1⋯vir xi1⋯xir = ∑ir=i1=i2=⋯=ir-1 xi1⋯xir = ∑1≤i1≤n xi1r = x1r+x2r+⋯+ xnr.$ $\square$

## 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.