Definition. \( S \) be a set. A matrix \( m : S \times S \to \{1, 2, \ldots, \infty \} \) is called a Coxeter matrix if it satisfies \[ m(s, s') = m(s', s); \tag{1.1} \] \[ m(s, s') = 1 \iff s = s'. \tag{1.2} \]
\(\textbf{Definition. }\). Let \( (W, S) \) be a Coxeter system and \( T = \{ wsw^{-1} : w \in W, s \in S \} \) its set of reflections. \textbf{Definition 2.1.1} Let \( u, w \in W \). Then \[ (i) \quad u \xrightarrow{t} w \quad \text{means that} \quad u^{-1}w = t \in T \quad \text{and} \quad \ell(u) < \ell(w). \] \[ (ii) \quad u \to w \quad \text{means that} \quad u \xrightarrow{t} w \quad \text{for some} \quad t \in T. \] \[ (iii) \quad u \leq w \quad \text{means that there exist} \quad w_i \in W \quad \text{such that} \] \[ u = u_0 \to u_1 \to \cdots \to u_{k-1} \to u_k = w. \]
Definition. Let \( W \) be a Coxeter group with a set of simple reflections \( S \). The Bruhat order on \( W \) is defined by \[ w \leq w' \iff l(w') - l(w) = \# \{ s \in S : w' < ws \}. \tag{2.1} \]
Proposition. The Bruhat order is a partial order on \( W \).\(\textbf{Proposition 10.10 (Deodhar)}\) If \( a_1, \ldots, a_K \) is a word comprised of integers, write \( a_1, \ldots, a_K \uparrow \) for the word rewritten in increasing order. For two words \( a_1, \ldots, a_K \) and \( b_1, \ldots, b_K \), write \( a_1, \ldots, a_K \preceq b_1, \ldots, b_K \) if \( a_i \leq b_i \) for all \( i \). Then \( u \leq v \) in \( S_n \) if and only if \( w(1) \cdots w(K) \preceq v(1) \cdots v(K) \) for \( K \in [n] \).
\(\textbf{Definition (Trivial representation)}.\) The trivial representation of a group \( G \) is the homomorphism \( \varphi : G \to \mathbb{C}^* \) given by \( \varphi(g) = 1 \) for all \( g \in G \).
Representation Theory of Finite Groups, Page 11
\(\textbf{Definition (Standard representation of \( S_n \)).}\) Define \( \varphi : S_n \to GL_n(\mathbb{C}) \) on basis elements by \( \varphi_{\sigma}(e_i) = e_{\sigma(i)} \). One obtains the matrix for \( \varphi_{\sigma} \) by permuting the rows of the identity matrix according to \( \sigma \). So, for instance, when \( n = 3 \) we have \[ \varphi_{(1 \; 2)} = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad \varphi_{(1 \; 2 \; 3)} = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}. \] Notice in Example 3.1.4 that \[ \varphi_{\sigma}(e_1 + e_2 + \cdots + e_n) = e_{\sigma(1)} + e_{\sigma(2)} + \cdots + e_{\sigma(n)} = e_1 + e_2 + \cdots + e_n \] where the last equality holds since \( \sigma \) is a permutation and addition is commutative. Thus \( \mathbb{C}(e_1 + \cdots + e_n) \) is invariant under all the \( \varphi_{\sigma} \) with \( \sigma \in S_3 \). This leads to the following definition.
Representation Theory of Finite Groups, Page 10
\(\textbf{Definition 13.6.}\) Let \( (\rho, V) \) be a representation of \( G \). Its character is the function \( \chi_{\rho} : G \to \mathbb{K} \) given by \[ \chi_{\rho}(g) = \text{tr}(\rho(g)). \]
\(\textbf{Lemma 2.7.}\) Any representation \( V \) of a finite group \( G \) can be given a \( G \)-invariant inner product, meaning that for any \( h \) in \( G \) and for any \( v_1, v_2 \) in \( V \), \( \langle hv_1, hv_2 \rangle = \langle v_1, v_2 \rangle \).
\(\textbf{Proof.}\) Let \( \langle \cdot, \cdot \rangle_* \) be any positive-definite Hermitian inner product on \( V \). Define a new Hermitian inner product on \( V \) in the following way: \[ \langle v_1, v_2 \rangle = \frac{1}{|G|} \sum_{g \in G} \langle gv_1, gv_2 \rangle_* \quad \text{for any } v_1, v_2 \in V. \] Now see that for any \( h \) in \( G \) and for any \( v_1, v_2 \) in \( V \), we have: \[ \begin{align} \langle hv_1, hv_2 \rangle &= \frac{1}{|G|} \sum_{g \in G} \langle g(hv_1), g(hv_2) \rangle_* \\ &= \frac{1}{|G|} \sum_{g \in G} \langle (gh)v_1, (gh)v_2 \rangle_* \\ &= \frac{1}{|G|} \sum_{g \in G} \langle gv_1, gv_2 \rangle_* \quad \text{because for every } g \in G, \; gh \text{ is in } G \text{ also} \\ &= \langle v_1, v_2 \rangle. \end{align} \]
Representations of the Symmetric Group
\(\textbf{Theorem 13.3 (Maschke's Theorem)}\) Let \( G \) be a finite group, and \( \mathbb{K} \) a field such that \( \text{char} \, \mathbb{K} \) does not divide the order of \( G \). Let \( (\rho, V) \) be a representation of \( G \). Then every \( G \)-invariant subspace has a \( G \)-invariant complement. In particular, \( (\rho, V) \) is semisimple.
From the Class Notes.
We define a Hermitian inner product on \(\mathbb{C}_{\text{class}}(G)\) by \[ (\alpha, \beta) = \frac{1}{|G|} \sum_{g \in G} \overline{\alpha(g)} \beta(g). \]
Representation Theory: A first class Page. 16
\(\textbf{Corollary 2.15.}\) A representation \( V \) is irreducible if and only if \( (\chi_V, \chi_V) = 1 \).
\(\textbf{Remark.}\) In fact, if \( V \cong V_1^{\oplus a_1} \oplus \cdots \oplus V_k^{\oplus a_k} \) as above, then \( (\chi_V, \chi_V) = \sum a_i^2 \). The multiplicities \( a_i \) can be calculated via
Representation Theory: A first class Page. 17