\documentclass[numbers=enddot,12pt,final,onecolumn,notitlepage]{scrartcl}% \usepackage[headsepline,footsepline,manualmark]{scrlayer-scrpage} \usepackage[all,cmtip]{xy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{framed} \usepackage{amsmath} \usepackage{comment} \usepackage{color} \usepackage{hyperref} \usepackage[sc]{mathpazo} \usepackage[T1]{fontenc} \usepackage{amsthm} %TCIDATA{OutputFilter=latex2.dll} %TCIDATA{Version=5.50.0.2960} %TCIDATA{LastRevised=Saturday, January 28, 2017 00:37:44} %TCIDATA{SuppressPackageManagement} %TCIDATA{} %TCIDATA{} %TCIDATA{BibliographyScheme=Manual} %BeginMSIPreambleData \providecommand{\U}{\protect\rule{.1in}{.1in}} %EndMSIPreambleData \theoremstyle{definition} \newtheorem{theo}{Theorem}[section] \newenvironment{theorem}[] {\begin{theo}[#1]\begin{leftbar}} {\end{leftbar}\end{theo}} \newtheorem{lem}[theo]{Lemma} \newenvironment{lemma}[] {\begin{lem}[#1]\begin{leftbar}} {\end{leftbar}\end{lem}} \newtheorem{prop}[theo]{Proposition} 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\newenvironment{example}[] {\begin{exmp}[#1]\begin{leftbar}} {\end{leftbar}\end{exmp}} \iffalse \newenvironment{proof}[Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \fi \newenvironment{verlong}{}{} \newenvironment{vershort}{}{} \newenvironment{noncompile}{}{} \excludecomment{verlong} \includecomment{vershort} \excludecomment{noncompile} \newcommand{\kk}{\mathbf{k}} \newcommand{\id}{\operatorname{id}} \newcommand{\ev}{\operatorname{ev}} \newcommand{\Comp}{\operatorname{Comp}} \newcommand{\bk}{\mathbf{k}} \let\sumnonlimits\sum \let\prodnonlimits\prod \let\bigcapnonlimits\bigcap \renewcommand{\sum}{\sumnonlimits\limits} \renewcommand{\prod}{\prodnonlimits\limits} \renewcommand{\bigcap}{\bigcapnonlimits\limits} \setlength\textheight{22.5cm} \setlength\textwidth{15cm} \ihead{Errata to Invariant Theory with Applications''} \ohead{\today} \begin{document} \begin{center} \textbf{Invariant Theory with Applications} \textit{Jan Draisma and Dion Gijswijt} \url{http://www.win.tue.nl/~jdraisma/teaching/invtheory0910/lecturenotes12.pdf} version of 7 December 2009 \textbf{Errata and addenda by Darij Grinberg} \bigskip \end{center} The following is a haphazard list of errors I found in \textquotedblleft Invariant Theory with Applications\textquotedblright\ by Jan Draisma and Dion Gijswijt. \begin{verlong} All page numbers given below are to be understood as page numbers in these notes. \end{verlong} \setcounter{section}{15} \section{Errata} \begin{itemize} \item \textbf{Page 5, \S 1.1:} Replace \textquotedblleft Clearly, the elements of $V^{\ast}$ are regular of degree\textquotedblright\ by \textquotedblleft Clearly, the elements of $V^{\ast}$ are regular functions and are homogeneous of degree\textquotedblright. \item \textbf{Page 7, \S 1.3:} \textquotedblleft discribed\textquotedblright% \ $\rightarrow$ \textquotedblleft described\textquotedblright. \item \textbf{Page 8, Example 1.3.2:} \textquotedblleft althought\textquotedblright\ $\rightarrow$ \textquotedblleft although\textquotedblright. \item \textbf{Page 8, Example 1.3.3:} \textquotedblleft with the same exponent\textquotedblright\ $\rightarrow$ \textquotedblleft with the same coefficient\textquotedblright. \item \textbf{Page 9, proof of Proposition 1.4.1:} Replace \textquotedblleft To each $c=(c_{1},\ldots,c_{n}\in\mathbb{C}^{n}$\textquotedblright\ by \textquotedblleft To each $c=\left( c_{1},\ldots,c_{n}\right) \in \mathbb{C}^{n}$\textquotedblright. \item \textbf{Page 9, proof of Proposition 1.4.1:} In (1.8), the entries in the last column should be $-c_{n},-c_{n-1},\ldots,-c_{2},-c_{1}$ (not $-c_{n},-c_{n-1},\ldots,c_{2},c_{1}$). \item \textbf{Page 9, proof of Proposition 1.4.1:} Replace \textquotedblleft shows that $\chi_{A_{c}}\left( t\right) =t^{n}+c_{n-1}t^{n-1}+\cdots +c_{1}t+c_{0}$\textquotedblright\ by \textquotedblleft shows that $\chi _{A_{c}}\left( t\right) =t^{n}+c_{1}t^{n-1}+\cdots+c_{n-1}t+c_{n}%$\textquotedblright. \item \textbf{Page 9, Exercise 1.4.2:} Replace \textquotedblleft that $\chi_{A_{c}}\left( t\right) =t^{n}+c_{n-1}t^{n-1}+\cdots+c_{1}t+c_{0}%$\textquotedblright\ by \textquotedblleft that $\chi_{A_{c}}\left( t\right) =t^{n}+c_{1}t^{n-1}+\cdots+c_{n-1}t+c_{n}$\textquotedblright. \item \textbf{Page 9, proof of Proposition 1.4.1:} In (1.9), replace \textquotedblleft$(s_{1}\left( A_{c}\right) ,s_{2}\left( A_{c}\right) ,\ldots,s_{n}\left( A_{c}\right)$\textquotedblright\ by \textquotedblleft% $\left( s_{1}\left( A_{c}\right) ,s_{2}\left( A_{c}\right) ,\ldots ,s_{n}\left( A_{c}\right) \right)$\textquotedblright. \item \textbf{Page 9, proof of Proposition 1.4.1:} Replace \textquotedblleft dense in $\mathcal{O}\left( \operatorname*{Mat}\nolimits_{n}\left( \mathbb{C}\right) \right)$\textquotedblright\ by \textquotedblleft dense in $\operatorname*{Mat}\nolimits_{n}\left( \mathbb{C}\right)$% \textquotedblright. (This mistake appears twice.) \item \textbf{Page 9, Exercise 1.4.3:} Replace \textquotedblleft nonzero eigenvalues\textquotedblright\ by \textquotedblleft eigenvalues\textquotedblright. \item \textbf{Page 10, Exercise 1.4.3:} Replace \textquotedblleft distinct and nonzero\textquotedblright\ by \textquotedblleft nonzero\textquotedblright. \item \textbf{Page 10, Exercise 1.4.3:} It might be worth noticing that \textquotedblleft the fact\textquotedblright\ you are mentioning about the Vandermonde determinant is a consequence of Lemma 2.2.4 below (using the well-known fact that the determinant of a square matrix equals the determinant of its transpose). \item \textbf{Page 15, Theorem 2.2.9:} You misspell \textquotedblleft Sylvester\textquotedblright\ as \textquotedblleft Sylverster\textquotedblright. \item \textbf{Page 15, proof of Theorem 2.2.9:} Remove the comma in \textquotedblleft Since, $\widetilde{A}$ contains\textquotedblright. \item \textbf{Page 15, proof of Theorem 2.2.9:} You write: \textquotedblleft it follows that $\operatorname*{Bez}\left( f\right)$ has rank $2k+r$\textquotedblright. How does this follow? I only see that $\operatorname*{Bez}\left( f\right)$ has rank $\leq2k+r$. \item \textbf{Page 20:} Replace \textquotedblleft every element $T\in U\otimes V$\textquotedblright\ by \textquotedblleft every element $t\in U\otimes V$\textquotedblright. \item \textbf{Page 20:} Replace \textquotedblleft for $T$ to zero\textquotedblright\ by \textquotedblleft for $t$ to zero\textquotedblright. \item \textbf{Page 21:} \textquotedblleft with of $k$% -tensors\textquotedblright\ $\rightarrow$ \textquotedblleft with $k$-tensors\textquotedblright. \item \textbf{Page 23:} \textquotedblleft so that the $v^{\alpha},\ \left\vert \alpha\right\vert =k$ a basis of $V$\textquotedblright\ should be \textquotedblleft so that the $v^{\alpha}$ with $\left\vert \alpha\right\vert =k$ form a basis of $S^{k}V$\textquotedblright. \item \textbf{Page 23:} Replace \textquotedblleft$\pi\left( v_{1}% \otimes\cdots v_{k}\right)$\textquotedblright\ by \textquotedblleft% $\pi\left( v_{1}\otimes\cdots\otimes v_{k}\right)$\textquotedblright. \item \textbf{Page 23, Exercise 3.0.13:} Replace \textquotedblleft$v\otimes v\cdots\otimes v$\textquotedblright\ by \textquotedblleft$v\otimes v\otimes\cdots\otimes v$\textquotedblright. \item \textbf{Page 24, Exercise 3.1.4:} You should require that at least one of $U$ and $V$ is finite-dimensional. \item \textbf{Page 24, Exercise 3.1.4:} Replace \textquotedblleft isomorhism\textquotedblright\ by \textquotedblleft isomorphism\textquotedblright. \item \textbf{Page 25:} Replace \textquotedblleft so that $g\left( hf\right) =\left( hg\right) f$\textquotedblright\ by \textquotedblleft so that $g\left( hf\right) =\left( gh\right) f$\textquotedblright. \item \textbf{Page 25, Example 4.0.8:} Replace \textquotedblleft$G$ module\textquotedblright\ by \textquotedblleft$G$-module\textquotedblright. \item \textbf{Page 26, Example 4.0.9:} Replace \textquotedblleft$G$ module\textquotedblright\ by \textquotedblleft$G$-module\textquotedblright. \item \textbf{Page 27, proof of Proposition 4.0.7:} \textquotedblleft$\left( v\mid v\right) =\sum_{g\in G}\left( gv\mid gv\right)$\textquotedblright% \ should be \textquotedblleft$\left( v\mid v\right) =\sum_{g\in G}\left( gv\mid gv\right) ^{\prime}$\textquotedblright. \item \textbf{Page 28, Lemma 4.1.1:} Replace \textquotedblleft$G$ modules\textquotedblright\ by \textquotedblleft$G$-modules\textquotedblright. \item \textbf{Page 28, \S 4.1:} \textquotedblleft of the isomorphism classes of $G$-modules\textquotedblright\ should be \textquotedblleft of the isomorphism classes of irreducible $G$-modules\textquotedblright. \item \textbf{Page 29, Exercise 4.1.2:} Remove the superscript \textquotedblleft$^{G}$\textquotedblright. \item \textbf{Page 31, Lemma 5.0.9:} \textquotedblleft Dixon's\textquotedblright\ $\rightarrow$ \textquotedblleft Dickson's\textquotedblright. \item \textbf{Page 32, proof of Hilbert's Basis Theorem:} \textquotedblleft Dixon's\textquotedblright\ $\rightarrow$ \textquotedblleft Dickson's\textquotedblright. \item \textbf{Page 32:} In (5.2), add a whitespace before \textquotedblleft for all $f\in V_{1}$\textquotedblright. \item \textbf{Page 32:} \textquotedblleft This a $G$-module morphism\textquotedblright\ $\rightarrow$ \textquotedblleft This is a $G$-module morphism\textquotedblright. \item \textbf{Page 33, Exercise 5.0.13:} \textquotedblleft with zero coefficient\textquotedblright\ should be \textquotedblleft with constant coefficient equal to $0$\textquotedblright. \item \textbf{Page 33, Exercise 5.0.13:} I am wondering whether you really mean \textquotedblleft subalgebra\textquotedblright\ here and not \textquotedblleft graded subalgebra\textquotedblright. \item \textbf{Page 34, proof of Theorem 5.1.1:} Replace \textquotedblleft% $\left\vert \beta\right\vert <=\left\vert G\right\vert$\textquotedblright\ by \textquotedblleft$\left\vert \beta\right\vert \leq\left\vert G\right\vert$\textquotedblright. \item \textbf{Page 34, proof of Theorem 5.1.1:} Replace \textquotedblleft% $p_{j}=\sum_{\left\vert \alpha\right\vert =j}f_{\alpha}z_{1}^{\alpha_{1}% }\cdots x_{n}^{\alpha_{n}}$\textquotedblright\ by \textquotedblleft$p_{j}% =\sum_{\left\vert \alpha\right\vert =j}f_{\alpha}z_{1}^{\alpha_{1}}\cdots z_{n}^{\alpha_{n}}$\textquotedblright. \item \textbf{Page 34, proof of Theorem 5.1.1:} You write: \textquotedblleft Recall that $p_{j}$ is a polynomial in $p_{1},\ldots,p_{\left\vert G\right\vert }$\textquotedblright. Did you actually prove this anywhere? (This is a particular case of the following fact: In the polynomial ring $\mathbb{C}\left[ x_{1},x_{2},\ldots,x_{n}\right]$, each $S_{n}$-invariant polynomial $f\in\mathbb{C}\left[ x_{1},x_{2},\ldots,x_{n}\right] ^{S_{n}}$ can be written as a polynomial in the Newton polynomials $p_{1},p_{2}% ,\ldots,p_{n}$.\ \ \ \ \footnote{The \textit{proof} of this fact is easy: By Theorem 2.1.1, it suffices to show that the $s_{1},s_{2},\ldots,s_{n}$ are polynomials in $p_{1},p_{2},\ldots,p_{n}$. In other words, it suffices to show that $s_{k}$ is a polynomial in $p_{1},p_{2},\ldots,p_{n}$ for each $k\in\left\{ 1,2,\ldots,n\right\}$. But this easily follows by strong induction over $k$ (indeed, (2.18) gives a way to write each $s_{k}$ for $k\in\left\{ 1,2,\ldots,n\right\}$ as a polynomial in $p_{1},p_{2}% ,\ldots,p_{n}$, provided that $s_{1},s_{2},\ldots,s_{k-1}$ have already been written in this form).} This is probably worth stating as an exercise in Chapter 2. \item \textbf{Page 37, proof of the weak Nullstellensatz:} Replace \textquotedblleft$f_{k,\xi}:=\left( x_{1},\ldots,x_{n-1},\xi\right)$\textquotedblright\ by \textquotedblleft$f_{k,\xi}:=f_{k}\left( x_{1}% ,\ldots,x_{n-1},\xi\right)$\textquotedblright. \item \textbf{Page 37, proof of the weak Nullstellensatz:} Replace all three \textquotedblleft$\sum_{i=1}^{k}$\textquotedblright\ signs by \textquotedblleft$\sum_{j=1}^{k}$\textquotedblright\ signs. \item \textbf{Page 37:} \textquotedblleft Nulstellensatz\textquotedblright% \ $\rightarrow$ \textquotedblleft Nullstellensatz\textquotedblright. \item \textbf{Page 39, proof of Theorem 6.1.10:} Replace the \textquotedblleft% $\sum_{i=1}^{k}$\textquotedblright\ sign by a \textquotedblleft$\sum_{j=1}% ^{k}$\textquotedblright\ sign. \item \textbf{Page 41, Lemma 6.2.6:} Replace \textquotedblleft from $\mathbb{C}\left[ Y\right]$ $\mathbb{C}\left[ X\right]$% \textquotedblright\ by \textquotedblleft from $\mathbb{C}\left[ Y\right]$ to $\mathbb{C}\left[ X\right]$\textquotedblright. \item \textbf{Page 41, proof of Lemma 6.2.8:} \textquotedblleft are a regular maps\textquotedblright\ $\rightarrow$ \textquotedblleft are regular maps\textquotedblright. \item \textbf{Page 42, Example 6.3.3:} Replace \textquotedblleft act on the $W$\textquotedblright\ by \textquotedblleft act on the vector space $W$\textquotedblright. \item \textbf{Page 43, Theorem 6.3.4:} In property 4, replace \textquotedblleft$\phi:Z\mapsto\mathbb{C}^{m}$\textquotedblright\ by \textquotedblleft$\phi:Z\rightarrow\mathbb{C}^{m}$\textquotedblright. \item \textbf{Page 43, proof of Theorem 6.3.4:} In the proof of property 3, replace \textquotedblleft$\phi:Z\mapsto U$\textquotedblright\ by \textquotedblleft$\phi:Z\rightarrow\mathbb{C}^{m}$\textquotedblright. \item \textbf{Page 47, proof of Theorem 7.0.14:} Replace \textquotedblleft Hence $w$ is in the null-cone $N_{V}$\textquotedblright\ by \textquotedblleft Hence $w$ is in the null-cone $N_{W}$\textquotedblright. \item \textbf{Page 49:} Replace \textquotedblleft Let $W\bigoplus _{d=0}^{\infty}W_{d}$ be a direct sum\textquotedblright\ by \textquotedblleft Let $W=\bigoplus_{d=0}^{\infty}W_{d}$ be a direct sum\textquotedblright. \item \textbf{Page 49:} In (8.1), replace \textquotedblleft$V$% \textquotedblright\ and \textquotedblleft$V_{d}$\textquotedblright\ by \textquotedblleft$W$\textquotedblright\ and \textquotedblleft$W_{d}%$\textquotedblright, respectively. \item \textbf{Page 49, Example 8.0.18:} Replace \textquotedblleft$H\left( \mathbb{C}\left[ x_{1},\ldots,x_{n}\right] \right)$\textquotedblright\ by \textquotedblleft$H\left( \mathbb{C}\left[ x_{1},\ldots,x_{n}\right] ,t\right)$\textquotedblright. \item \textbf{Page 50, Theorem 8.1.1:} Replace \textquotedblleft of a finite group\textquotedblright\ by \textquotedblleft of a finite group $G$% \textquotedblright. \item \textbf{Page 50, proof of Theorem 8.1.1:} In (8.5), replace \textquotedblleft$\operatorname*{tr}\left( L_{d}\left( g\right) \right)$\textquotedblright\ by \textquotedblleft$t^{d}\operatorname*{tr}\left( L_{d}\left( g\right) \right)$\textquotedblright. \item \textbf{Page 50, proof of Theorem 8.1.1:} Replace \textquotedblleft lets fix\textquotedblright\ by \textquotedblleft let's fix\textquotedblright. \item \textbf{Page 50, proof of Theorem 8.1.1:} Replace \textquotedblleft the inner sum $\sum_{d=0}^{\infty}\operatorname*{tr}\left( L_{d}\left( g\right) \right)$\textquotedblright\ by \textquotedblleft the inner sum $\sum _{d=0}^{\infty}t^{d}\operatorname*{tr}\left( L_{d}\left( g\right) \right)$\textquotedblright. \item \textbf{Page 50, proof of Theorem 8.1.1:} You write: \textquotedblleft Pick a basis $x_{1},\ldots,x_{n}$ of $V^{\ast}$ that is a system of eigenvectors for $L_{1}\left( g\right)$\textquotedblright. It is worth justifying why such a basis exists. (Namely, you are using the apocryphal theorem from linear algebra that says that if $U$ is a finite-dimensional $\mathbb{C}$-vector space, and if $\alpha$ is an element of $\operatorname*{GL}\left( U\right)$ having finite order, then $\alpha$ is diagonalizable. You are applying this theorem to $U=V^{\ast}$ and $\alpha=L_{1}\left( g\right)$, which is allowed because the element $L_{1}\left( g\right)$ of $\operatorname*{GL}\left( V^{\ast}\right)$ has finite order (since the element $g$ of $G$ has finite order). This is not a difficult argument, but I don't think it is obvious enough to be entirely left to the reader.) \item \textbf{Page 50, proof of Theorem 8.1.1:} Replace \textquotedblleft for a system\textquotedblright\ by \textquotedblleft form a system\textquotedblright. \item \textbf{Page 50, proof of Theorem 8.1.1:} On the first line of the computation (8.7), replace \textquotedblleft$\left( 1+\lambda_{n}% t+\lambda_{n}t^{2}+\cdots\right)$\textquotedblright\ by \textquotedblleft% $\left( 1+\lambda_{n}t+\lambda_{n}^{2}t^{2}+\cdots\right)$% \textquotedblright. \item \textbf{Page 50, proof of Theorem 8.1.1:} On the first line of the computation (8.8), replace \textquotedblleft$\operatorname*{tr}\left( L_{d}\left( g\right) \right)$\textquotedblright\ by \textquotedblleft% $t^{d}\operatorname*{tr}\left( L_{d}\left( g\right) \right)$% \textquotedblright. \item \textbf{Page 50, proof of Theorem 8.1.1:} On the third line of the computation (8.8), replace \textquotedblleft$\det(I-\rho\left( g\right) t$\textquotedblright\ by \textquotedblleft$\det\left( I-\rho\left( g\right) t\right)$\textquotedblright. \item \textbf{Page 51, \S 8.2:} Replace \textquotedblleft which $u$ and $v$, differ\textquotedblright\ by \textquotedblleft which $u$ and $v$ differ\textquotedblright. \item \textbf{Page 51, \S 8.1:} It is worth pointing out that you use the word \textquotedblleft code\textquotedblright\ to mean \textquotedblleft linear code\textquotedblright. \item \textbf{Page 52:} \textquotedblleft Furhermore\textquotedblright% \ $\rightarrow$ \textquotedblleft Furthermore\textquotedblright. \item \textbf{Page 53, Theorem 8.2.6:} Replace \textquotedblleft$\left( x^{4}-y^{4}\right)$\textquotedblright\ by \textquotedblleft$\left( x^{4}-y^{4}\right) ^{4}$\textquotedblright. \item \textbf{Page 57, Example 9.1.8:} Replace \textquotedblleft$\prod _{k}\prod_{l}$\textquotedblright\ by \textquotedblleft$\sum_{k}\sum_{l}%$\textquotedblright. \item \textbf{Page 59, \S 9.2:} Replace \textquotedblleft Consider the map $\lambda:G\rightarrow\operatorname*{GL}[\mathbb{C}\left[ x_{ij},1/\det\left( x\right) \right] )$\textquotedblright\ by \textquotedblleft Consider the map $\lambda:G\rightarrow\operatorname*{GL}\left( \mathbb{C}\left[ x_{ij}% ,1/\det\left( x\right) \right] \right)$\textquotedblright. \item \textbf{Page 65, Exercise 10.0.12:} Replace \textquotedblleft larger enough\textquotedblright\ by \textquotedblleft large enough\textquotedblright. \item \textbf{Page 65, proof of Proposition 10.0.13:} Replace \textquotedblleft standard basis $\mathbb{C}^{2}$\textquotedblright\ by \textquotedblleft standard basis of $\mathbb{C}^{2}$\textquotedblright. \item \textbf{Page 65, proof of Proposition 10.0.13:} Replace \textquotedblleft induced basis of $S^{d}\left( V\right)$\textquotedblright% \ by \textquotedblleft induced basis of $S^{k}\left( V\right)$% \textquotedblright. \item \textbf{Page 65, proof of Proposition 10.0.13:} Replace \textquotedblleft$\sum_{i}d\left( \lambda\right) x^{i}y^{k-i}$% \textquotedblright\ by \textquotedblleft$\sum_{i}d_{i}\left( \lambda\right) x^{i}y^{k-i}$\textquotedblright. \item \textbf{Page 65, proof of Proposition 10.0.13:} You claim that \textquotedblleft$d_{0}$ and every $d_{i}$ with $c_{i}\neq0$ are nonzero polynomials with $\lambda$\textquotedblright. I would suggest explaining why they are nonzero. (Namely, the polynomial $d_{0}$ is nonzero because $d_{0}=\sum_{i}c_{i}\lambda^{i}y^{k}$ (and because not all $c_{i}$ are $0$); meanwhile, the polynomials $d_{i}$ with $c_{i}\neq0$ are nonzero because they satisfy $d_{i}\left( 0\right) =c_{i}\neq0$.) \item \textbf{Page 66, proof of Proposition 10.0.13:} Replace \textquotedblleft Then for every $i$ the vector $\mu^{k}\left( \begin{array} [c]{cc}% \mu & 0\\ 0 & \mu^{-1}% \end{array} \right) u=\sum_{i}\lambda^{i}c_{i}x^{i}y^{k-i}$ belongs to $U$% \textquotedblright\ by \textquotedblleft Then for every $\mu\in\left\{ \mu_{0},\mu_{1},\ldots,\mu_{k}\right\}$ the vector $\mu^{k}\left( \begin{array} [c]{cc}% \mu & 0\\ 0 & \mu^{-1}% \end{array} \right) u=\sum_{i}\lambda^{i}c_{i}x^{i}y^{k-i}$ (with $\lambda=\mu^{2}$) belongs to $U$\textquotedblright. \item \textbf{Page 66, proof of Proposition 10.0.13:} In (10.4), replace \textquotedblleft$S^{d}\left( \operatorname*{End}\left( S^{2}\left( V\right) \oplus\mathbb{C}\right) \right)$\textquotedblright\ by \textquotedblleft$S^{d}\left( S^{2}\left( V\right) \oplus\mathbb{C}\right)$\textquotedblright. \item \textbf{Page 66, Exercise 10.0.14:} Replace \textquotedblleft% $\operatorname*{SL}\nolimits_{2}\left( \mathbb{C}\right)$ module\textquotedblright\ by \textquotedblleft$\operatorname*{SL}% \nolimits_{2}\left( \mathbb{C}\right)$-module\textquotedblright. \item \textbf{Page 68, \S 11.1:} Replace \textquotedblleft it identifies the space $\operatorname*{End}\left( V^{\otimes k}\right) ^{S_{k}}$ with $\left( \operatorname*{End}\left( V\right) ^{\otimes k}\right) ^{S_{k}}$ of symmetric tensors\textquotedblright\ by \textquotedblleft it identifies the space $\operatorname*{End}\left( V^{\otimes k}\right) ^{S_{k}}$ with the space $\left( \operatorname*{End}\left( V\right) ^{\otimes k}\right) ^{S_{k}}$ of symmetric tensors\textquotedblright. \item \textbf{Page 68, \S 11.1:} Replace \textquotedblleft Applying the following theorem to $H=S_{n}$\textquotedblright\ by \textquotedblleft Applying the following theorem to $H=S_{k}$\textquotedblright. \item \textbf{Page 69, proof of Theorem 11.2.1:} \textquotedblleft represations\textquotedblright\ $\rightarrow$ \textquotedblleft representations\textquotedblright. \item \textbf{Page 69, proof of Theorem 11.2.1:} You write: \textquotedblleft By complete reducibility, the map $\left( \left( U^{\ast}\right) ^{\otimes d}\right) ^{G}\rightarrow\left( S^{d}U^{\ast}\right) ^{G}$ is surjective\textquotedblright. Actually, you don't need to use complete reducibility here: The projection map $\pi:\left( U^{\ast}\right) ^{\otimes d}\rightarrow S^{d}U^{\ast },\ \ \ \ \ \ \ \ \ \ u_{1}\otimes u_{2}\otimes\cdots\otimes u_{d}\mapsto u_{1}u_{2}\cdots u_{d}%$ has a $G$-equivariant section -- namely, the linear map% $\psi:S^{d}U^{\ast}\rightarrow\left( U^{\ast}\right) ^{\otimes d}% ,\ \ \ \ \ \ \ \ \ \ u_{1}u_{2}\cdots u_{d}\mapsto\dfrac{1}{d!}\sum_{\sigma\in S_{d}}u_{\sigma\left( 1\right) }\otimes u_{\sigma\left( 2\right) }% \otimes\cdots\otimes u_{\sigma\left( d\right) }.$ Hence, the restriction $\left( \left( U^{\ast}\right) ^{\otimes d}\right) ^{G}\rightarrow\left( S^{d}U^{\ast}\right) ^{G}$ of the map $\pi$ to the $G$-invariants has a section as well (namely, the restriction of the section $\psi:S^{d}U^{\ast}\rightarrow\left( U^{\ast}\right) ^{\otimes d}$ to the $G$-invariants). Therefore, this restriction is surjective. I like this argument more not just because it avoids the use of complete reducibility, but also because it is more general (it works for any subgroup $G$ of $\operatorname*{GL}\nolimits_{n}$, including those for which the representations involved fail to be completely reducible). \item \textbf{Page 70, proof of Theorem 11.2.1:} \textquotedblleft If $d=k$\textquotedblright\ should be \textquotedblleft If $k=d-k$% \textquotedblright. \item \textbf{Page 74:} \textquotedblleft a fix a stochastic\textquotedblright% \ $\rightarrow$ \textquotedblleft we fix a stochastic\textquotedblright. \item \textbf{Page 75, \S 12.3:} Replace \textquotedblleft formal linear combinations of the alphabet $V$\textquotedblright\ by \textquotedblleft formal linear combinations of the alphabet $B$\textquotedblright. \item \textbf{Page 75, \S 12.3:} Replace \textquotedblleft Next we define a polynomial map $\psi_{T}:\operatorname*{rep}\left( T\right) \rightarrow \bigotimes_{p\in\operatorname*{leaf}\left( T\right) }$\textquotedblright\ by \textquotedblleft Next we define a polynomial map $\Psi_{T}% :\operatorname*{rep}\left( T\right) \rightarrow\bigotimes_{p\in \operatorname*{leaf}\left( T\right) }V_{p}$\textquotedblright. (There were two typos here: \textquotedblleft$\psi_{T}$\textquotedblright\ should be \textquotedblleft$\Psi_{T}$\textquotedblright, and the \textquotedblleft% $V_{p}$\textquotedblright\ was missing.) \item \textbf{Page 76, \S 12.3:} I suppose that \textquotedblleft% $\bigotimes_{p\in\operatorname*{leaf}\left( T\right) f\left( p\right) }%$\textquotedblright\ should be \textquotedblleft$\bigotimes_{p\in \operatorname*{leaf}\left( T\right) }f\left( p\right)$\textquotedblright. \end{itemize} \end{document}