\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=Monday, October 19, 2015 11:25:20} %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} \newenvironment{proposition}[] {\begin{prop}[#1]\begin{leftbar}} {\end{leftbar}\end{prop}} \newtheorem{defi}[theo]{Definition} \newenvironment{definition}[] {\begin{defi}[#1]\begin{leftbar}} {\end{leftbar}\end{defi}} \newtheorem{remk}[theo]{Remark} \newenvironment{remark}[] {\begin{remk}[#1]\begin{leftbar}} {\end{leftbar}\end{remk}} \newtheorem{coro}[theo]{Corollary} \newenvironment{corollary}[] {\begin{coro}[#1]\begin{leftbar}} {\end{leftbar}\end{coro}} \newtheorem{conv}[theo]{Convention} \newenvironment{condition}[] {\begin{conv}[#1]\begin{leftbar}} {\end{leftbar}\end{conv}} \newtheorem{quest}[theo]{Question} \newenvironment{algorithm}[] {\begin{quest}[#1]\begin{leftbar}} {\end{leftbar}\end{quest}} \newtheorem{warn}[theo]{Warning} \newenvironment{conclusion}[] {\begin{warn}[#1]\begin{leftbar}} {\end{leftbar}\end{warn}} \newtheorem{conj}[theo]{Conjecture} \newenvironment{conjecture}[] {\begin{conj}[#1]\begin{leftbar}} {\end{leftbar}\end{conj}} \newtheorem{exmp}[theo]{Example} \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}} \newcommand{\Nplus}{\mathbb{N}_{+}} \newcommand{\NN}{\mathbb{N}} \let\sumnonlimits\sum \let\prodnonlimits\prod \renewcommand{\sum}{\sumnonlimits\limits} \renewcommand{\prod}{\prodnonlimits\limits} \setlength\textheight{22.5cm} \setlength\textwidth{15cm} \ihead{Errata to On the straightening law...''} \ohead{\today} \begin{document} \begin{center} \textbf{On the straightening law for minors of a matrix} \textit{Richard G. Swan} \texttt{\href{http://www.math.uchicago.edu/~swan/strLaw.pdf}{http://www.math.uchicago.edu/\symbol{126}% swan/strLaw.pdf}} version of 18 February 2003 \textbf{Errata and addenda by Darij Grinberg} \bigskip \end{center} I will refer to the results appearing in the preprint \textquotedblleft On the straightening law for minors of a matrix\textquotedblright\ by the numbers under which they appear in this preprint (specifically, in its version of 18 February 2003, published on \newline% \texttt{\href{http://www.math.uchicago.edu/~swan/strLaw.pdf}{http://www.math.uchicago.edu/\symbol{126}% swan/strLaw.pdf}}). \setcounter{section}{10} \section{Errata} \begin{itemize} \item \textbf{Page 1:} The name \textquotedblleft Doubilet\textquotedblright% \ is misspelt as \textquotedblleft Doubillet\textquotedblright\ twice (once in the abstract, and once again in \S 1)., \item \textbf{Page 2, proof of Lemma 2.3:} Replace \textquotedblleft etc. ,each\textquotedblright\ by \textquotedblleft etc., each\textquotedblright. \item \textbf{Page 2, proof of Theorem 2.6:} The \textquotedblleft$\sigma$\textquotedblright\ comes out of the blue in the proof, as there is no $\sigma$ in the statement of the theorem. To clarify where it comes from, I suggest adding the following sentence at the beginning of the proof: \textquotedblleft By Corollary 2.5, we can WLOG assume that $X$ is a permutation matrix $P\left( \sigma^{-1}\right)$ for some $\sigma \in\mathcal{S}_{n}$.\textquotedblright. \item \textbf{Page 3, \S 3:} Replace \textquotedblleft say that S is good\textquotedblright\ by \textquotedblleft say that $S$ is good\textquotedblright. (I have replaced the text-\textquotedblleft S\textquotedblright\ by a math-\textquotedblleft$S$\textquotedblright\ here.) \item \textbf{Page 3, Theorem 3.1:} It would be helpful to clarify that some of the $\left( A_{i},B_{i}\right)$ may be equal. \item \textbf{Page 3, proof of Theorem 3.1:} Replace \textquotedblleft% $B_{i}\leq B$,and\textquotedblright\ by \textquotedblleft$B_{i}\leq B$, and\textquotedblright. \item \textbf{Page 3, proof of Theorem 3.1:} Replace \textquotedblleft all other terms\textquotedblright\ by \textquotedblleft all other nonzero terms\textquotedblright. \item \textbf{Page 4, proof of Theorem 4.1:} This proof is confusing due to some of its parts being out of order. The determinant $Y\left( P\mid Q\right)$ is not well-defined before the sets $I$ and $J$ are totally ordered; I even would not call $Y$ a \textquotedblleft square matrix\textquotedblright\ until the indexing set $I$ for its rows and the indexing set $J$ for its columns have been identified. I suggest modifying the proof as follows: \begin{itemize} \item At the beginning of the proof, add the following sentence: \textquotedblleft We WLOG assume that $\left\vert S^{\prime}\right\vert =\left\vert T^{\prime}\right\vert$ and $\left\vert S^{\prime\prime }\right\vert =\left\vert T^{\prime\prime}\right\vert$, as otherwise the left hand side is $0$.\textquotedblright. \item Move the second paragraph of the proof (the paragraph that begins with \textquotedblleft Order $I$ by setting\textquotedblright\ and ends with \textquotedblleft and similarly for subsets of $J$\textquotedblright) to before the sentence that begins with \textquotedblleft Define a square matrix $Y$\textquotedblright. \item Before the sentence that begins with \textquotedblleft Define a square matrix $Y$\textquotedblright, add the following sentence: \textquotedblleft Now the sets $I$ and $J$ are totally ordered. Let $k=\left\vert I\right\vert =\left\vert J\right\vert$. Then, we have order-preserving bijections $I\rightarrow\left\{ 1,2,\ldots,k\right\}$ and $J\rightarrow\left\{ 1,2,\ldots,k\right\}$. Use these bijections to identify $I$ and $J$ with $\left\{ 1,2,\ldots,k\right\}$. For every subset $Q$ of $\left\{ 1,2,\ldots,k\right\}$, we will write $\widetilde{Q}$ for the complement $\left\{ 1,2,\ldots,k\right\} -Q$.\textquotedblright. \end{itemize} Once this is done, $Y$ actually becomes a honest square matrix (of size $k\times k$). \item \textbf{Page 4, proof of Theorem 4.1:} Replace \textquotedblleft$\left( S^{\prime}\mid T^{\prime}\right) \left( S^{\prime\prime}\mid T^{\prime \prime}\right) =Y\left\{ I^{\prime}\mid J^{\prime}\right\}$% \textquotedblright\ by \textquotedblleft$\left( S^{\prime}\mid T^{\prime }\right) \left( S^{\prime\prime}\mid T^{\prime\prime}\right) =\pm Y\left\{ I^{\prime}\mid J^{\prime}\right\}$\textquotedblright. (At least I don't see a reason why the $\pm$ must always be a $+$. Maybe it is?) \item \textbf{Page 5, proof of Lemma 4.2:} Add \textquotedblleft WLOG assume that $\left\vert K\right\vert =\left\vert I^{\prime}\right\vert$ (since otherwise, $\varphi\left( K\right) \neq S^{\prime}$ is obvious).\textquotedblright\ after the first sentence of the proof. \item \textbf{Page 5, proof of Lemma 4.2:} Remove \textquotedblleft, $S^{\prime}=\left\{ s_{1}<\cdotsi_{\nu}$\textquotedblright\ goes a bit too fast for me. I suggest adding a few details, e.g., as follows: \textquotedblleft Recall that $\varphi$ is injective on $I^{\prime}$ and on $K$. Thus, from $I^{\prime}=\left\{ i_{1}<\cdotsi_{\nu}$\textquotedblright. \item \textbf{Page 5, \S 5:} A nitpick: Replace \textquotedblleft whose entries are indeterminates\textquotedblright\ by \textquotedblleft whose entries are distinct indeterminates\textquotedblright. \item \textbf{Page 5, Theorem 5.3:} In this theorem, you probably want to state that two standard monomials which only differ by factors of the form $\left( \varnothing,\varnothing\right)$ are considered to be identical. (You can always add $\left( \varnothing,\varnothing\right)$ to the end of a standard monomial without changing the value of this monomial.) \item \textbf{Page 6, proof of Theorem 5.3:} Replace all appearances of \textquotedblleft leading form\textquotedblright\ by \textquotedblleft leading term\textquotedblright\ (or is \textquotedblleft leading form\textquotedblright\ really a synonym for \textquotedblleft leading term\textquotedblright?). \item \textbf{Page 6, proof of Theorem 5.3:} Add a period before \textquotedblleft Similarly, if\textquotedblright. \item \textbf{Page 6, proof of Theorem 5.3:} After \textquotedblleft$z\left( B\right) =z_{b_{1}}^{\left( 1\right) }\cdots z_{b_{p}}^{\left( p\right) }$\textquotedblright, add \textquotedblleft(if $N\geq\left\vert A\right\vert =\left\vert B\right\vert$)\textquotedblright. \item \textbf{Page 6, References:} \textquotedblleft Desarmenian\textquotedblright\ $\rightarrow$ \textquotedblleft Desarmenien\textquotedblright\ in . \end{itemize} \end{document}