$\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}$ We will now compute the irreps of $\GL_n(\CC)$ through [[Representations of SL|representations of $\SL_n(\CC)$]]. We observe that $\GL_n(\CC) = (\CC^\times\times \SL_n(\CC))/\mu_n$ where $\mu_n$ consists of terms $(\omega^{-1},\omega \id)\in \CC^\times\times\SL_n$; the covering homomorphism is via $\CC^\times \times \SL_n(\CC)\to \GL_n(\CC): (z,A)\mapsto zA$, and turn our attention to $\CC^\times \times \SL_n(\CC)$. If $n = 1$, representation of $\CC^\times$ are completely reducible, with irreps $\chi_N$ all one-dimensional via $\chi_N(z) = z^N$. (Obviously, the corresponding Lie algebra representation is $z\mapsto Nz$). In general, $\CC^\times \times \SL_n$ representations are still completely reducible, and all of the form $L_{\lambda, N} = \chi_N\otimes L_\lambda$ (this is clear). In order to factor through $\GL_n$, they need to annihilate $(\omega^{-1}, \omega \id)$. The $\CC^\times$ part acts as $\omega^{-N}$ on $L_{\lambda, N}$, while $\omega \id$ acts as $\omega^\abs{\lambda}$. >[!proof]- Short calculation >In the lie algebra with $\omega = e^{\frac{2\pi ik}{n}}$, this maps to e.g.$\left(-\frac{2\pi i k N}{n}, +\frac{2\pi i k}{n} \underbrace{\begin{pmatrix}1&&&\\&1&&\\&&\ddots&\\&&&-n+1\end{pmatrix}}_h\right).$$h$ is in the CSA of $\sl_n$, and thus acts by a dot product against the weight. This dot product is $\lambda(h) = \frac{2\pi i k}{n} \abs{\lambda}$ ($\lambda_n = 0$ for $\SL_n$-reps). Thus a representation factors iff $N = nm_n + \sum_{i}^{n-1} \lambda_i$ for some $m_n$, and we restrict to this case from now on. # Results Working in reductive $\gl_n$ now, $\hh = \CC^n$ has an additional axis (which we append to the back). Highest weights now take the form$(m_1+\dots+m_n,\dots,m_n)$n.b. the last term is no longer $0$. We rename $\lambda$ to this. Observe now that $N = \abs{\lambda}$, i.e. the $\CC^\times$ component acts as $z\mapsto z^{\abs{\lambda}}$; this is not surprising, because $\id\in \gl_n$ lives in root space $(1,\dots,1)$. The fundamental representations are $\wedge^mV$. $\wedge^nV$ is no longer trivial, but the ==**determinant character**== with highest weight $\omega_n = (1,\dots,1)$. The highest weight of a finite-dimensional representation is $\lambda = \sum_{i = 1}^n m_i \omega_i$ where $m_i\geq 0$ for $i < n$ and $m_n\in \ZZ$, and $L_\lambda$ is found inside $\bigotimes_i (\wedge^i V)^{\otimes m_i}$. >[!idea] One-dimensional determinant character admits "negative tensor powers" >Allow $\chi^{\otimes k} = (\chi^*)^{\otimes -k}$ if $k < 0$. When $m_n\geq 0$, the $\lambda$ lives naturally inside some $V^{\otimes N}$, while if $m_n\leq 0$, it does not (the action of $\omega \id$ on some subrep of $V^{\otimes N}$ should always have positive determinant). >[!idea] Polynomial representations >These representations are called ==**polynomial**== because their matrix coefficients are polynomial functions of the entries in $\GL_n(\CC)$; consequently they extend by continuity to representations of $\End_n(\CC)$. Note that any irrep is a polynomial rep tensored with a non-positive power of $\wedge^nV$.