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title: Review Session Notes

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# Review Session Notes

For Monica Vazirani's course, with Jennifer Brown & Anne Dranowski

**Notation:**
* $G$ and $H$ denote groups, and $g,h$ denote group elements
* $S$ denotes a set
* $S_n$ denotes the symmetric group on $n$ elements.
* $GL_d$ denotes the group of $d\times d$ invertible matrices.
* $V,W$ are modules/representations.
* $k$ is a field.

## Exercises

We recommend Chapter 1 of *The Symmetric Group* by Bruce E. Sagan as a reference. A (digital) copy is available in the library in Sococo. (You will need to be on the IAS VPN.) 

1. Suppose that $X: G \to Mat_d$ is a map satisfying (i) that $X(ab) = X(a)X(b)$ for all $a,b$ in $G$ and (ii) $X(e) = 1_d$ where $e \in G$ is the identity and $1_d \in Mat_d$ is the identity matrix.

    Show that $X(a) \in GL_d$ for all $a\in G$. What's $X(a^{-1})$ in terms of $X(a)$?  <!-- $X(a^{-1}) = X(a)^{-1}$.  -->
    
2. Check that $V$ is a $G$-module if and only if there's a multiplication map $G\times V \to V$ such that
    1. $gv \in V$,
    2. $g(av + bw) = a(gv) + b(gw)$,
    3. $(gh)v = g(hv)$, and
    4. $1v = v$, for all $a,b\in\mathbb C$, $g\in G$, $v,w\in V$.

1. Check that the definitions of a module and a representation are equivalent.

1. Given a set $S$ with an action of $G$, construct a $G$-module of dimension $|S|$.

1. Check that the orthogonal complement of a submodule with respect to an invariant inner-product is also a submodule.

1. Check that the kernel and image of a a $G$-module homomorphism are submodules. Use this to show that $G$-homomorphism of irreducible representations ("irreps") are either isomorphisms or trivial.

1. If $V,W$ are $G,H$-modules, respectively, then $V\boxtimes W$ is calledthe tensor product representation of $G\times H$. The the action is given by $(g,h)(v\otimes w) = gv\otimes hw$. Check that this is indeed a $G\times H$ representation.

1. Let $G = S_3$ and $G\supseteq H = \langle (2,3)\rangle$.
    1. Calculate the matrix form of the $G$-module $\text{Ind}_H^G(1)$. 
    1. This is the matrix of a permutation representation! Work out the matrix form of the coset representation $k[G/H]$ of $G$. 

## Further Exercises

1. (Sagan 1) An inversion of a permutation $\pi = [x_1 \, x_2 \, \dots \, x_n] \in S_n$ (1-line notation) is a pair $x_i,x_j$ such that $i < j$ but $x_i> x_j$. Let $\text{inv}(\pi)$ be the number of inversions of $\pi$. 
   1. Show that if $\pi$ can be written as a product of $k$ transpositions then the $k \equiv \text{inv}(\pi)\bmod 2$
   1. Check that $(-1)^\pi$ is well-defined.

1. (Sagan 2) Let $S$ be a $G$-set. Every $s\in S$ has *stabilizer* $G_s = \{g\in G: gs = s\}$, and *orbit* $O_s = \{gs : g \in G\}$. 
   1. Check that $G_s$ is a subgroup of $G$. 
   1. Find a bijection between $G/G_s$ and $O_s$.
   1. Show that $|G|/|G_s| = |O_s|$ and work out the formula for this number in the case that $G = S_n$.

