Last update: 19 August 2013
Homework Problem 1.60 Let $A$ be a finite dimensional commutative algebra over $C\text{.}$ Show that all irreducible representations of $A$ are one dimensional.
Homework Problem 1.61 Let $A$ be a finite dimensional semisimple algebra over $C\text{.}$ Show that $\text{dim}\hspace{0.17em}A={\sum}_{\lambda \in \stackrel{\u02c6}{A}}{d}_{\lambda}$ where ${d}_{\lambda}=\text{dim}\hspace{0.17em}{V}^{\lambda}$ is the dimension of the irreducible $A\text{-module}$ indexed by $\lambda \in \stackrel{\u02c6}{A}\text{.}$
I. | Basics |
II. | Characters and Symmetric Functions |
III. | Construction of Irreducible Modules using Young Symmetrizers. |
We won’t do the orthogonal representations of the symmetric group, but we will do them for the Hecke algebras later. The following references will prove useful.
For the orthogonal representations, consult
Definition 2.1 The symmetric group on $m$ letters, ${\mathcal{S}}_{m},$ is the group of permutations of the set of integers $[1,m]$ with product defined by composition.
We shall use several equivalent notations for an element $\pi \in {\mathcal{S}}_{m}\text{.}$
The symmetric group ${\mathcal{S}}_{m}$ also has a presentation defined by taking generators $\{{s}_{1},{s}_{2},{s}_{m-1}\}$ subject to the relations $$\begin{array}{ccc}\left(S1\right)& {s}_{i}{s}_{j}={s}_{j}{s}_{i}& |i-j|\ge 2\\ \left(S2\right)& {s}_{i}{s}_{i+1}{s}_{i}={s}_{i+1}{s}_{i}{s}_{i+1}& 1\le i\le m-2\\ \left(S3\right)& {s}_{i}^{2}=1& 1\le i\le m-1\end{array}$$ These generators are called simple transpositions. The transposition ${s}_{i}$ corresponds to the permutation $(i\hspace{0.17em}i+1)$ written in cycle notation.
We have seen that the trace property implies that characters of a group are constant on conjugacy classes. Furthermore, the irreducible characters are indexed by the same set which indexes the set of conjugacy classes. We thus wish to find the conjugacy classes of the symmetric group ${\mathcal{S}}_{m}\text{.}$
Let ${\gamma}_{r}=\left(1\hspace{0.17em}2\hspace{0.17em}\dots \hspace{0.17em}r\right)\in {\mathcal{S}}_{m},$ which is represented by the diagram $$\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n1\n2\n3\n4\n\cdots \nr-1\nr\nr+1\n\cdots \nm\n\n$$ For each finite sequence of positive integers $\mu =({\mu}_{1},{\mu}_{2},\dots ,{\mu}_{k}),$ define ${\gamma}_{\mu}={\gamma}_{{\mu}_{1}}\times {\gamma}_{{\mu}_{2}}\times \cdots \times {\gamma}_{{\mu}_{k}}\in {\mathcal{S}}_{m}$ where $m={\sum}_{i=1}^{k}{\mu}_{i}\text{.}$ Here ${\gamma}_{i}\times {\gamma}_{j}$ means the element of ${\mathcal{S}}_{i+1}$ obtained by placing the diagram ${\gamma}_{j}$ to the right of the diagram for ${\gamma}_{i}\text{.}$ For example, if $\mu =(3,2,1),$ then ${\gamma}_{\mu}\in {\mathcal{S}}_{6}$ is the element $${\gamma}_{\mu}=\begin{array}{c}\stackrel{\stackrel{{\gamma}_{3}}{\u23de}}{\n\n\n\n\n\n\n\n\n\n\n\n\n}\hspace{0.17em}\stackrel{\stackrel{{\gamma}_{2}}{\u23de}}{\n\n\n\n\n\n\n\n\n\n}\hspace{0.17em}\stackrel{\stackrel{{\gamma}_{1}}{\u23de}}{\n\n\n\n\n\n\n}\end{array}$$
Next we define a procedure that associates to a permutation $\pi \in {\mathcal{S}}_{m}$ a sequence of positive integers ${\mu}_{1},{\mu}_{2},{\mu}_{3},\dots ,{\mu}_{k}$ such that $$\genfrac{}{}{0ex}{}{{\mu}_{1}\ge {\mu}_{2}\ge {\mu}_{3}\ge \cdots {\mu}_{k}>0}{{\sum}_{i=1}^{k}{\mu}_{i}=m}$$ Such a sequence of integers is called a partition of $m,$ and we write $\mu \u22a2m\text{.}$
To produce a partition from $\pi \in {\mathcal{S}}_{m},$ overlay they identity diagram on the diagram of $\pi \text{.}$ Each edge of the original diagram of $\pi $ now belongs to exactly one of several disjoint cycles in this new graph. Let ${\mu}_{i}$ be the number of edges of the original diagram of $\pi $ which lie in the $i\text{th}$ cycle. For example, given $\pi =\left(1\hspace{0.17em}4\right)\left(3\hspace{0.17em}6\hspace{0.17em}5\right)\in {\mathcal{S}}_{6},$ we obtain a graph $$\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n$$ from which we obtain the integers 2, 1, and 3. Arranging these in decreasing order gives the partition $(3,2,1)$ of the number 6.
Define the cycle type of the permutation $\pi $ to be the partition $\mu \left(\pi \right)$ obtained by the above procedure. Observe that given a partition $\mu ,$ the permutation ${\gamma}_{\mu}$ defined above has cycle type $\mu \text{.}$
Proposition 2.2 Let $\sigma \in {\mathcal{S}}_{m}\text{.}$ Then σ is conjugate to ${\gamma}_{\mu \left(\sigma \right)},$ where $\mu \left(\sigma \right)$ is the cycle type of $\sigma \text{.}$
Proof. | |
Write $\mu =\mu \left(\sigma \right)=({\mu}_{1},{\mu}_{2},\dots ,{\mu}_{k})\text{.}$ Since $\sigma $ has cycle type $\mu ,$ the diagram of $\sigma $ is partitioned into disjoint cycles of lengths ${\mu}_{i}\text{.}$ It is merely a question of conjugating $\sigma $ by the permutation that arranges theses cycles in the order in which they appear in ${\gamma}_{\mu}\text{.}$ We obtain this permutation as follows. Recall that the vertices of $\sigma $ in the top row are labeled from left to right by the integers $[1,m]\text{.}$ Let ${c}_{i,1}$ denote the label of the leftmost vertex of the top row belonging to the cycle of length ${\mu}_{i}\text{.}$ To assign values to ${c}_{i,j}$ for $2\le j\le {\mu}_{i},$ we traverse the cycle of length ${\mu}_{i}$ in the diagram of $\pi $ by proceeding along the edge of $\pi $ from top to bottom and then jumping from the bottom row to the corresponding vertex in the top row. For $2\le j\le {\mu}_{i},$ let ${c}_{i,j}$ be the label of the $j\text{th}$ vertex visited in the top row of $\pi \text{.}$ Define $\rho $ to be the permutation diagram where the $\ell \text{th}$ vertex in the top row is adjacent to the ${c}_{i,j}\text{th}$ vertex in the bottom row where $\ell ={\mu}_{1}+{\mu}_{2}+\cdots +{\mu}_{i-1}+j$ and $1\le j\le {\mu}_{i}\text{.}$ Observe that this expression uniquely defines $i$ and $j$ and the set of ${c}_{i,j}$ is in one to one correspondence with $[1,m],$ so $\rho $ is a well-defined permutation diagram. Moreover, ${\rho}^{-1}$ is obtained by inverting the diagram of $\rho ,$ so that the ${c}_{i,j}\text{th}$ vertex in the top row is adjacent to the $\ell ={\mu}_{1}+{\mu}_{2}+\cdots +{\mu}_{i-1}+j$ vertex of the bottom $\rho \text{.}$ Note that each edge in sigma belongs to exactly one of the cycles of $\pi \text{;}$ hence, a given edge runs from the vertex in the top row labeled by ${c}_{i,j}$ to the vertex in the bottom row below the vertex labeled by ${c}_{i,j+1}$ (where we take ${c}_{i,{\mu}_{i}+1}={c}_{i,1}\text{).}$ In the composition of the product, therefore, we pass from the vertex in the top row or $\rho $ labeled by $\ell ={\mu}_{1}+\cdots +{\mu}_{i-1}+j$ to the ${c}_{i,j}\text{th}$ vertex in the top row of $\sigma ,$ whence to the ${c}_{i,j+1}$ vertex of the top row of ${\rho}^{-1},$ which by definition is adjacent to the $(\ell +1)\text{st}$ vertex of the bottom row for $1\le j\le {\mu}_{i},$ or to the vertex ${\mu}_{1}+{\mu}_{2}+\cdots +{\mu}_{i-1}+1$ if $j={\mu}_{i}\text{.}$ This results in precisely the diagram of ${\gamma}_{\mu}\text{.}$ $\square $ |
As an example, consider the permutation $\pi =\left(1\hspace{0.17em}4\right)\left(3\hspace{0.17em}6\hspace{0.17em}5\right)$ with cycle type $(3,2,1),$ above. Then this process results in the product $$\begin{array}{ccccccc}\rho \pi {\rho}^{-1}& =& \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n& =& \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n& =& {\gamma}_{\mu}\end{array}$$
Since the ${\gamma}_{\mu}$ for distinct partitions of $m$ are not conjugate, we have the following corollary.
Proposition 2.3 The conjugacy classes of ${\mathcal{S}}_{m}$ are indexed by the set of partitions of $m\text{.}$
As usual, we will denote the set of partitions of $m$ by $\stackrel{\u02c6}{{\mathcal{S}}_{m}}\text{.}$ This is not standard, but agrees with our previously defined notation.
We next introduce some notation and language concerning partitions. Let $\mu =({\mu}_{1},{\mu}_{2},\dots ,{\mu}_{k})$ be a partition of $m\text{.}$ The individual terms of the sequence ${\mu}_{i}$ are called the parts of $\mu \text{.}$ We may use exponential notation to indicate that a certain part is repeated, e.g. $(4,2,2,1,1,1)$ may be written $(4,{2}^{2},{1}^{3})\text{.}$ We identify a partition $\mu =({\mu}_{1},{\mu}_{2},\cdots ,{\mu}_{k})$ with all partitions obtained by padding $\mu $ with zeros, i.e. $\mu =({\mu}_{1},{\mu}_{2},\dots ,{\mu}_{k},{0}^{n})\text{.}$ The length of $\mu ,$ denoted $\ell \left(\mu \right)$ is the number of nonzero parts. We will work with partitions of different integers simultaneously, and sometimes will denote the sum of the parts of a partition by $\left|\mu \right|,$ so $\mu \u22a2\left|\mu \right|\text{.}$
As always, we work with pictures; for each partition $\mu ,$ we construct the associated Ferrar’s diagram of $\mu ,$ which is a frame consisting of $m$ boxes placed as follows. Working top to bottom and left-justified, we place ${\mu}_{1}$ boxes in the first row, ${\mu}_{2}$ boxes in the second row, and so on. The boxes are then labeled using matrix notation, i.e. the $(i,j)$ box is the $j\text{th}$ box in the $i\text{th}$ row from the top. Hence, the partition $(6,{4}^{2},2,{1}^{4})\u22a221$ is represented by the Ferrar’s diagram $$\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n$$
The conjugate of a partition $\mu $ is the partition $\stackrel{\sim}{\mu}=({\stackrel{\sim}{\mu}}_{1},{\stackrel{\sim}{\mu}}_{2},\cdots {\stackrel{\sim}{\mu}}_{r}),$ where ${\stackrel{\sim}{\mu}}_{j}=\left\{{\mu}_{i}\hspace{0.17em}\right|\hspace{0.17em}{\mu}_{i}\ge j\}\text{.}$ That is, ${\stackrel{\sim}{\mu}}_{j}$ is the length of the $j\text{th}$ column of $\mu $ and hence $\stackrel{\sim}{\mu}=m\text{.}$ Moreover, the Ferrar’s diagram for $\stackrel{\sim}{\mu}$ is the transpose of the Ferrar’s diagram for $\mu \text{.}$
Let ${\alpha}_{i}={\mu}_{i}-i$ and ${\beta}_{j}={\stackrel{\sim}{\mu}}_{j}-j$ for $1\le i\le \ell \left(\mu \right)$ and $1\le j\le \ell \left(\stackrel{\sim}{\mu}\right)\text{.}$ Looking at the Ferrars diagram for $\mu ,$ we see that ${\alpha}_{i}$ is the number of boxes strictly to the right of the diagonal in the $i\text{th}$ row and ${\beta}_{j}$ is the number of boxes strictly below the diagonal in the $j\text{th}$ column. The Frobenius notation partition, Frobenius notation for for $\mu $ is $({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{\ell \left(\mu \right)}\hspace{0.17em}|\hspace{0.17em}{\beta}_{1},{\beta}_{2},\dots ,{\beta}_{\ell \left(\stackrel{\sim}{\mu}\right)})\text{.}$
Let us now identify some representations of the symmetric group. Let ${V}^{\left(m\right)}:\u2102\left[{\mathcal{S}}_{m}\right]\to {M}_{1}\left(\u2102\right)=\u2102$ be the trivial representation ${V}^{\left(m\right)}\left(\sigma \right)=1$ for all $\sigma \in {\mathcal{S}}_{m}\text{.}$ Clearly ${V}^{\left(m\right)}$ is irreducible as it is not zero and has dimension one. The character of this representation ${\chi}^{\left(m\right)}:\u2102\left[{\mathcal{S}}_{m}\right]\to \u2102$ is also defined by ${\chi}^{\left(m\right)}\left(\sigma \right)=1\text{.}$ Hence, the minimal central idempotent corresponding to this representation is given by $${z}_{\left(m\right)}=\frac{1}{\left|G\right|}\sum _{\sigma \in {\mathcal{S}}_{m}}\text{dim}\hspace{0.17em}{V}^{\left(m\right)}{\chi}^{\left(m\right)}\left({\sigma}^{-1}\right)\sigma =\frac{1}{m!}\sum _{\sigma \in {\mathcal{S}}_{m}}\sigma $$
Let us find all one dimensional representations of ${\mathcal{S}}_{m},$ which necessarily are irreducible. Suppose $\psi :\u2102\left[{\mathcal{S}}_{m}\right]\to \u2102$ is a representation. Recall the presentation of the symmetric group ${\mathcal{S}}_{m}$ defined on generators $\{{s}_{1},{s}_{2},\dots ,{s}_{m-1}\}$ defined above. $$1=\psi \left(1\right)=\psi \left({s}_{i}^{2}\right)=\psi {\left({s}_{i}\right)}^{2}$$ so $\psi \left({s}_{i}\right)=\pm 1$ for each $i\text{.}$ With an eye toward relation S2, however, $$\begin{array}{ccc}\psi \left({s}_{i}{s}_{i+1}{s}_{i}\right)& =& {(\pm 1)}^{2}\psi \left({s}_{i+1}\right)=\psi \left({s}_{i+1}\right)\\ \psi \left({s}_{i+1}{s}_{i}{s}_{i+1}\right)& =& {(\pm 1)}^{2}\psi \left({s}_{i}\right)=\psi \left({s}_{i}\right)\text{.}\end{array}$$ Hence $\psi \left({s}_{i}\right)=\psi \left({s}_{j}\right)$ for all $i$ and $j\text{.}$ If this common value is 1, we have the trivial representation ${V}^{\left(m\right)}$ described above.
Hence, the representation of ${\mathcal{S}}_{m},$ ${V}^{\left({1}^{m}\right)}:\u2102\left[{\mathcal{S}}_{m}\right]\to \u2102,$ defined by ${V}^{\left({1}^{m}\right)}\left({s}_{i}\right)=-1,$ is the only other one-dimensional representation besides the trivial representation ${V}^{\left(m\right)}\text{.}$ The character ${\chi}^{\left({1}^{m}\right)}$ is called the sign character of ${\mathcal{S}}_{m}\text{.}$
Finally, we introduce a related algebra.
Definition 2.4 Let $q$ be an indeterminate over $\u2102$ and let $\u2102\left(q\right)$ denote the rational functions in $q\text{.}$ The (Iwahori) Hecke algebra, ${\mathscr{H}}_{m},$ is the associative algebra with identity over $\u2102\left(q\right)$ generated by the elements $\{{g}_{1},{g}_{2},\dots ,{g}_{m-1}\}$ subject to the relations $$\begin{array}{ccc}\left(H1\right)& {g}_{i}{g}_{j}={g}_{j}{g}_{i}& \text{for}\hspace{0.17em}|i-j|\ge 2\\ \left(H2\right)& {g}_{i}{g}_{i+1}{g}_{i}={g}_{i+1}{g}_{i}{g}_{i+1}& 1\le u\le m-2\\ \left(H3\right)& {g}_{i}^{2}=(q-1){g}_{i}+q& 1\le i\le m-1\end{array}$$
Homework Problem 2.5 Find all one dimensional representations of ${\mathscr{H}}_{m}\text{.}$
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.