DecompositionOfAModuleUsingOrthogonalIdempotents 2005-04-25 17:25:22 2005-04-27 14:11:10 Application orthogonal idempotents of the group ring % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newtheorem{thm}{Theorem} \newtheorem{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem*{lemma}{Lemma} \newtheorem{cor}{Corollary} \theoremstyle{definition} \newtheorem{exa}{Example} % Some sets \newcommand{\Nats}{\mathbb{N}} \newcommand{\Ints}{\mathbb{Z}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} \newcommand{\Rats}{\mathbb{Q}} \newcommand{\Gal}{\operatorname{Gal}} \newcommand{\Cl}{\operatorname{Cl}} Let $K$ be a field and let $G$ be a finite abelian group. For simplicity, we will assume that the characteristic of $K$ does not divide the order of $G$. Let $\varphi_1,\ldots, \varphi_n$ be a complete set (up to equivalence) of distinct \PMlinkname{irreducible}{GroupRepresentation} (linear) representations of $G$ over $K$, so that $\varphi_i$ is a homomorphism: $$\varphi_i\colon G \longrightarrow \operatorname{GL}(n_i,K)$$ where $n_i$ is the degree of the representation $\varphi_i$ and $\sum_i n_i=|G|$. Let $\chi_1,\ldots,\chi_n$ be the irreducible characters attached to the $\varphi_i$, i.e. the function $\chi_i\colon G \to K$ is defined by $$\chi_i(g)=\text{Trace}(\varphi_i(g)).$$ Notice, however, that in general the map $\chi_i$ is not a homomorphism from the group into either the additive or multiplicative group of $K$. We define a system of primitive orthogonal idempotents of the group ring $K[G]$, one for each $\chi_i$, by: $${\bf 1}_{\chi_i}=\frac{1}{|G|}\sum_{g\in G} \chi_i(g^{-1})g\in K[G]$$ so that $\sum_{i}{\bf 1}_{\chi_i}=1\in K$ and ${\bf 1}_{\chi_i}\cdot {\bf 1}_{\chi j}=\delta_{ij}$ where $\delta_{ij}$ is the Kronecker delta function. We define the $\chi_i$ component of $K[G]$to be the ideal $K[G]_{\chi_i}={\bf 1}_{\chi_i}\cdot K[G]$. Notice that $V_i=K[G]_{\chi_i}$ is a finite dimensional $K$-vector space, on which $G$ acts. Thus, the representation of $G$ afforded by the $K[G]$-module $V_i$, call it $\varphi$, must be one of the representations $\varphi_j$ defined above. Comparing the trace, one concludes that $\varphi=\varphi_i$ and $V_i=K[G]_{\chi_i}$ is a vector space of dimension $n_i$. In particular, there is a decomposition: $$K[G]=\oplus_\chi K[G]_\chi.$$ If $k\in K[G]$ then by the previous decomposition, we can write: $$k=\sum_\chi k_\chi$$ where $k_\chi \in K[G]_\chi$. Notice that the representations $\varphi_i$ can be retrieved as: $$\varphi_i\colon G \longrightarrow \operatorname{GL(K[G]_{\chi_i})}.$$ \begin{lemma} Let $M$ be a $K[G]$-module and define submodules $M_\chi={\bf 1}_{\chi}\cdot M$, for each irreducible character $\chi$. Then: \begin{enumerate} \item There is a decomposition $M=\oplus_\chi M_\chi$. \item The group $K[G]$ acts on $M_\chi$ via $K[G]_\chi$. In other words, if $k\in K[G]$, with $k=\sum_\chi k_\chi$ then: $$k\cdot m = k_\chi \cdot m, \text{ for all } m\in M_\chi.$$ \item The representation $\varphi$ of $G$ afforded by the $K$-vector space $M_{\chi_i}$ is, up to equivalence, a number of copies of $\varphi_i$, i.e. $$\varphi=\varphi_i\oplus \ldots \oplus \varphi_i=\varphi_i^{\oplus r}$$ for some integer $r\geq 0$. In other words, $M_{\chi_i}$ is the submodule consisting of the sum of all $K[G]$-submodules of $M$ isomorphic to $K[G]_{\chi_i}$. \item Suppose that $M$, $N$ and $R$ are $K[G]$-modules which fit in the short exact sequence: $$0\longrightarrow R \longrightarrow M \longrightarrow N \longrightarrow 0$$ where every map above is a $K[G]$-module homomorphism, i.e. each map is a $K$-homomorphism which is compatible with the action of $G$. Then, the exact sequence above yields an exact sequence of $\chi$ components: $$0\longrightarrow R_\chi \longrightarrow M_\chi \longrightarrow N_\chi \longrightarrow 0$$ for every irreducible character $\chi$. \end{enumerate} \end{lemma}