$\require{physics}\newcommand{\cbrt}[1]{\sqrt[3]{#1}}\newcommand{\sgn}{\text{sgn}}\newcommand{\ii}[1]{\textit{#1}}\newcommand{\eps}{\varepsilon}\newcommand{\EE}{\mathbb E}\newcommand{\PP}{\mathbb P}\newcommand{\Var}{\mathrm{Var}}\newcommand{\Cov}{\mathrm{Cov}}\newcommand{\pperp}{\perp\kern-6pt\perp}\newcommand{\LL}{\mathcal{L}}\newcommand{\pa}{\partial}\newcommand{\AAA}{\mathscr{A}}\newcommand{\BBB}{\mathscr{B}}\newcommand{\CCC}{\mathscr{C}}\newcommand{\DDD}{\mathscr{D}}\newcommand{\EEE}{\mathscr{E}}\newcommand{\FFF}{\mathscr{F}}\newcommand{\WFF}{\widetilde{\FFF}}\newcommand{\GGG}{\mathscr{G}}\newcommand{\HHH}{\mathscr{H}}\newcommand{\PPP}{\mathscr{P}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathbb{H}}\DeclareMathOperator{\ess}{ess}\newcommand{\CC}{\mathbb C}\newcommand{\FF}{\mathbb F}\newcommand{\NN}{\mathbb N}\newcommand{\QQ}{\mathbb Q}\newcommand{\RR}{\mathbb R}\newcommand{\ZZ}{\mathbb Z}\newcommand{\KK}{\mathbb K}\newcommand{\SSS}{\mathbb S}\newcommand{\II}{\mathbb I}\newcommand{\conj}[1]{\overline{#1}}\DeclareMathOperator{\cis}{cis}\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}\DeclareMathOperator*{\range}{range}\DeclareMathOperator*{\nul}{null}\DeclareMathOperator*{\Tr}{Tr}\DeclareMathOperator*{\tr}{Tr}\newcommand{\id}{1\!\!1}\newcommand{\Id}{1\!\!1}\newcommand{\der}{\ \mathrm {d}}\newcommand{\Zc}[1]{\ZZ / #1 \ZZ}\newcommand{\Zm}[1]{\left(\ZZ / #1 \ZZ\right)^\times}\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\End}{End}\newcommand{\GL}{\mathbb{GL}}\newcommand{\SL}{\mathbb{SL}}\newcommand{\SO}{\mathbb{SO}}\newcommand{\OO}{\mathbb{O}}\newcommand{\SU}{\mathbb{SU}}\newcommand{\U}{\mathbb{U}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\Cl}{\mathrm{Cl}}\newcommand{\gr}{\mathrm{gr}}\newcommand{\gl}{\mathfrak{gl}}\newcommand{\sl}{\mathfrak{sl}}\newcommand{\so}{\mathfrak{so}}\newcommand{\su}{\mathfrak{su}}\newcommand{\sp}{\mathfrak{sp}}\newcommand{\uu}{\mathfrak{u}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\hh}{\mathfrak{h}}\DeclareMathOperator{\Ad}{Ad}\DeclareMathOperator{\ad}{ad}\DeclareMathOperator{\Rad}{Rad}\DeclareMathOperator{\im}{im}\renewcommand{\BB}{\mathcal{B}}\newcommand{\HH}{\mathcal{H}}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Mat}{Mat}\DeclareMathOperator{\span}{span}\DeclareMathOperator{\proj}{proj}$ >[!theorem] Frobenius Character Formula >The character value of $\chi_\lambda(\sigma)$ is the coefficient of $x_1^{\lambda_1 + N - 1}\dots x_N^{\lambda_N}$ in the polynomial >$\prod_{i < j}(x_i - x_j) \cdot \prod_i (x^i_1+\dots x^i_n)^{m_i}$ # Proof >[!idea]- Weyl Character Formula >$\chi_\lambda = \frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda + \rho)}}{\Delta},\qquad \Delta = \sum_{w\in W}(-1)^{\ell(w)} e^{w\rho}.$ By the [[Weyl Character Formula]], the characters of the representations of $\GL_n$ are given by$s_\lambda(x_1,\dots, x_n) = \frac{\sum_{s\in S_n} \det(s) x^{\lambda_1 + N - 1}_{s(1)}\dots x^{\lambda_n}_{s(n)}}{\prod_{i < j} (x_i - x_j)} = \frac{\det(x_i^{\lambda_j + N - j})}{\prod_{i < j}(x_i - x_j)}$We call $s_\lambda$ a ==**Schur Polynomial**==. > [!idea] Why? > The Weyl group of $\sl_n$ is $S_n$, and the length $\ell(w)$ of an element just describes the minimum number of adjacent transpositions necessary to create $w$, whose parity is obviously its sign. What does $e^{w(\lambda + \rho)}(x_1,\dots, x_n)$ mean. Well, $x_1,\dots, x_n$ are actually supposed to be lie algebra elements. This by definition should act as $x_i^{w(\lambda + \rho)_i}$, which is $x_1^{\lambda_{w(i)} + \lambda_{w(i)}}$. But you can just invert the permutation, so we're fine. >[!example] $S^mV$ >$s_{(m)}(x_1,\dots, x_n) = \sum_{1\leq j_1\leq\dots\leq j_m\leq n} x_{j_1}\dots x_{j_m} = h_m(x_1,\dots,x_n)$where $h_m$ is the $m$-th ==**complete symmetric function.**== >[!example] $\wedge^mV$ >$s_{(1,\dots,1)}(x_1,\dots, x_n) = \sum_{1< j_1<\dots< j_m< n} x_{j_1}\dots x_{j_m} = e_m(x_1,\dots,x_n)$where $e_m$ is the $m$-th ==**elementary symmetric function.**== Consider the element $x\otimes \sigma\in \End(V^{\otimes N})$, where $x = \diag(x_1,\dots, x_n)\in \End(V)$ and $\sigma\in S_N$ is a permutation (this acts by applying $x$ to each $V$, then permuting the $Vs with $\sigma$). Suppose $\sigma$ has $m_i$ cycles of length $i$. Then,$\Tr\big\vert_{V^{\otimes N}} (x\otimes \sigma) = \prod_i(x^i_1+\dots+x^i_n)^{m_i}.$(This is literally by definition of trace). On the other hand, using Schur-Weyl duality, we have big brain decomposition go$\Tr\big\vert_{V^{\otimes N}} (x\otimes \sigma) = \sum_\lambda \chi_\lambda(\sigma) s_\lambda(x)$where $\chi_\lambda(\rho) = \Tr\big\vert_{\pi_\lambda}(\sigma)$ is the character of $\pi_\lambda$ in $S_N$. Equating, multiplying by $\prod_{i < j}$, and extracting coefficients, we obtain the theorem.