\documentclass{beamer}
\usepackage{amsmath}
\usepackage{bookman}
\usetheme{Madrid}

\title{Lecture 4-12: Induced characters and representations}
\date{April 12, 2024}
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}
\vskip .1in
Given a representation $V$ of a group $G$, one can clearly restrict to a subgroup $H$, whose character is just the restriction $\chi_H$ to $H$ of the character $\chi$ of $V$.  There is an operation called {\sl induction} going the other way, from characters of $H$ to characters of $G$, which provides a very useful way of obtaining representations of larger groups from those of smaller ones. 
\end{frame}
\begin{frame}
\vskip .1in
Given a class function $c_H$ on a subgroup $H$ of a group $G$, how could one use it to produce a class function $c_G$ on $G$?  To make sure that $c_G$ really is a class function, one needs to average over conjugacy classes in $G$, but the original function $c_H$ is defined only on $H$.  One is fairly naturally led to
\begin{alertblock}{Definition: DF p. 849, Corollary 12}
We define $c_G=$Ind$_H^G c_H$, the function induced from $c_H$, via $c_G(g) =\frac 1 {|H|} \sum\limits_{x\in G\atop x^{-1}gx\in H} c_H(x^{-1}gx)$, where $|H|$ denotes the order of $H$.  Equivalently, letting $g_1,\ldots,g_m$ to be a set of representatives of the left cosets of $H$ in $G$, we take $c_G(g) = \sum\limits_{g_i^{-1}xg_i\in H} c_H(g_i^{-1}xg_i)$.  
\end{alertblock}
Clearly $c_G=$Ind$_H^G c_H$ depends linearly on $c_H$.
\end{frame}
\begin{frame}
\vskip .1in
If $c_H$ is the character $\chi$ of a representation of $H$, then $c_G$ is also the character of a representation.  This follows from the next result, in which $(\cdot,\cdot)$ denotes the standard Hermitian inner product on characters defined earlier.
\begin{alertblock}{Theorem:  Frobenius reciprocity}
With notation as above, we have $(c_G,\rho) = (c_H,\rho_H)$ for all characters $\rho$ of $G$.
\end{alertblock}
\end{frame}
\begin{frame}
\vskip .1in
\begin{proof}
We have $(c_G,\rho) = \frac 1 {|G|}\sum\limits_{g\in G} c_G(g)\overline{\rho(g)} = \frac 1 {|G|}\frac 1 {|H|}\sum\limits_{g\in G}\sum\limits_{x\in G\atop x^{-1}gx\in H}c_H(x^{-1}gx)\overline{\rho(x^{-1}gx)}$.  In this last sum every term $c_H(h)\overline{\rho(h)}$ appears exactly $|G|$ times for each fixed $h\in H$, whence the sum equals $(c_H,\rho_H)=(\rho_H,c_H)$, as claimed.
\end{proof}
\vskip .1in
We express this last result by saying that \alert{induction from $H$ to $G$ is the left adjoint of restriction from $G$ to $H$}:  we move from the left to the right side of the inner product by replacing an induced representation by a restricted one.
\end{frame}
\begin{frame}
\vskip .1in
If $c_H$ is the character of an irreducible representation $V$ of $H$, then every irreducible representation $W$ of $G$ decomposes as a direct sum of irreducible representations of $H$, including $V$ some nonnegative number $n_W$ of times, whence $c_G$ is the character of the direct sum of $n_W$ copies of $W$ as $W$ runs over the irreducible representations of $G$.  If $c_H$ is the character of an arbitrary representation of $H$ then by linearity $c_G$ is again the character of a representation.
\end{frame}
\begin{frame}
\vskip .1in
It is also possible to construct the induced representations directly, as follows (DF, p. 893).  Given a representation $V$ of $H$, form the tensor product $W=\mathbb CG\otimes_{\mathbb CH} V$, regarding $\mathbb CG$ as a left module over itself and a right module over $\mathbb CH$, so that $xy\otimes v=x\otimes yv$ in $W$ if $x\in\mathbb CG,y\in\mathbb CH,v\in V$, and $\mathbb CG$ acts on $W$ by left multiplication on the left factor $\mathbb CG$.  Then the degree of $W$ is $\frac {|G|} {|H|}$ times the degree of $V$; more generally, any class function $c_G$ on $G$ induced by a class function $c_H$ on $H$ is such that $c_G(1) = \frac {|G|} {|H|}c_H(1)$.
\end{frame}
\begin{frame}
\vskip .1in
As an example, take $G=S_3$ and let $H$ be the cyclic subgroup generated by a transposition.  Inducing the trivial character from $H$ to $G$, we get a character taking the value 3 at the identity $e$, 1 on a transposition, and 0 on a 3-cycle; taking the square length of this character, we find that it is the sum of two irreducible characters.  Subtracting off the trivial character (which occurs in it by Frobenius reciprocity) we get the character denoted by $\chi_r$ in a previous lecture, taking the value 2 at $e$, 0 on a transposition, and $-1$ on a 3-cycle. 
\end{frame}
\begin{frame}
\vskip .1in
Thus if we had never heard of the representation with this character, we could reconstruct it by induction.  A similar calculation shows that inducing the nontrivial character from $H$ to $G$ gives the character with value 3 at $e, -1$ on a transposition, and 0 on a 3-cycle; subtracting off the character $\chi_r$ just constructed, we recover the character of the sign representation, which is 1 on $e$ and a 3-cycle and $-1$ on a transposition.
\end{frame}
\begin{frame}
\vskip .1in
We can also get interesting information by inducing class functions that are not characters.  As an example, start with the cyclic subgroup $T$ generated by a 3-cycle in the alternating group $A_4$ on four letters.  Take each of the one-dimensional characters of $T$, subtract off the trivial character, and induce the resulting class function to $A_4$.  You get $\chi_1-\chi_1,\chi_2-\chi_1,\chi_3-\chi_1$, where the $\chi_i$ range over the three one-dimensional characters of $A_4$ (constructed previously; here $\chi_1$ is the trivial character).  What is going on in both of these examples is that there is a subgroup $C$ of $G$, equal to $H$ in the first example and $T$ in the second, which is such that $g^{-1}Cg\cap C = 1$ for any $g\in G$ with $g\notin C$.  
 \end{frame}
\begin{frame}
\vskip .1in
More generally, let $G$ be a transitive subgroup of the symmetric group $S_n$ (acting on the index set $\{1,\ldots,n\}$ with a single orbit) such that only the identity in $G$ fixes as many as two indices among $1,\ldots,n$.  Let $H$ be the stabilizer  in $G$ of any index, say of 1.  The hypothesis on $G$ implies that $H\cap g^{-1}Hg = 1$ whenever $g\notin H$.  A famous result of Frobenius then asserts that \alert{the set of permutations in $G$ fixing no index, together with the identity, is a normal subgroup of $G$}; this is called the {\sl Frobenius kernel} while the subgroup $H$ is called the {\sl Frobenius complement}.  You will prove this in homework next week, using the theory of induced characters.
\end{frame}
\begin{frame}
\vskip .1in
As a final example, it is not difficult to check that if you induce the character of either $V_{n-1}$ or $V_{n-1}'$ of the Clifford group $G_{n-1}$ for $n$ even, you get the character of $V_n$ on $G_n$.
\end{frame}
\end{document}