\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=Thursday, February 13, 2020 20:40:57} %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}} \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 Determinants, Paths, and Plane Partitions''} \ohead{\today} \begin{document} \begin{center} \textbf{\href{http://people.brandeis.edu/~gessel/homepage/papers/pp.pdf}{\textbf{Determinants, Paths, and Plane Partitions}}} \textit{Ira M. Gessel, X. V. Viennot} (1989 preprint) version of 28 August 2000 \textbf{Errata by Darij Grinberg} \bigskip \end{center} \setcounter{section}{5} \section*{Errata} The following are my comments on specific places in the preprint \textquotedblleft% \textbf{\href{http://people.brandeis.edu/~gessel/homepage/papers/pp.pdf}{\textbf{Determinants, Paths, and Plane Partitions}}}\textquotedblright\ by Ira M. Gessel and X. V. Viennot (in its version of 28 August 2000). I have read only parts of the preprint. \begin{itemize} \item \textbf{page 2, proof of Theorem 1:} Replace the big \textquotedblleft% $\bigcup\limits_{\pi\in S_{k}}$\textquotedblright\ sign (in the second displayed equation of this proof) by a \textquotedblleft$\bigsqcup \limits_{\pi\in S_{k}}$\textquotedblright\ sign (which stands for an external disjoint union). In fact, the sets $\mathsf{P}\left( \mathbf{u},\pi\left( \mathbf{v}\right) \right) -\mathsf{N}\left( \mathbf{u},\pi\left( \mathbf{v}\right) \right)$ can have nonempty intersection for different permutations $\pi\in S_{k}$ when some of the $v_{i}$'s are equal. Thus we must take a disjoint union in order to ensure that each $k$-path in it \textquotedblleft knows\textquotedblright\ which $\pi$ it comes from. \item \textbf{page 2, proof of Theorem 1:} Before \textquotedblleft Then properties (i), (ii), and (iii) are easily verified\textquotedblright, I would add the following sentence: \textquotedblleft We then define $\mathbf{A}% ^{\ast}$ (that is, the image of $\mathbf{A}$ under our bijection) as the $k$-path $\left( A_{1}^{\ast},A_{2}^{\ast},\ldots,A_{k}^{\ast}\right) \in\mathsf{P}\left( \mathbf{u},\sigma\left( \mathbf{v}\right) \right)$, where $\sigma=\pi\circ\left( i,j\right)$.\textquotedblright\ (This should clarify which permutation $\mathbf{A}^{\ast}$ corresponds to when some of the $v_{i}$ are equal.) \item \textbf{page 2:} You write: \textquotedblleft Let us say that a pair $\left( \mathbf{u},\mathbf{v}\right)$ of $k$-vertices is \textit{nonpermutable} if $N\left( \mathbf{u},\pi\left( \mathbf{v}\right) \right)$ is empty\textquotedblright\ etc.. Here, \textquotedblleft% $N$\textquotedblright\ should be \textquotedblleft$\mathsf{N}$% \textquotedblright. \item \textbf{page 3:} On the first line of this page, replace \textquotedblleft for $i>1$\textquotedblright\ by \textquotedblleft for $i\in\left\{ 2,3,\ldots,\ell\left( \lambda\right) \right\}$% \textquotedblright\ (since sufficiently large $i$ would otherwise have to satisfy $0=0+1$). Also, this isn't how I would define a skew-hook. Your definition forces the skew hook to start in row $1$, which is unlike the standard definition that is used in the Murnaghan-Nakayama rule. \item \textbf{page 3:} On the first line of this page, replace \textquotedblleft\textit{skew hook}\textquotedblright\ by \textquotedblleft% \textit{skew-hook}\textquotedblright\ (since you later use the hyphenated version). \item \textbf{page 3:} In \textquotedblleft The plane partition $\left( p_{ij}\right)$ is \textit{row-strict} if (3.2) is replaced by $p_{ij}% >p_{i,j+1}$ and column-strictness is defined similarly\textquotedblright, replace \textquotedblleft(3.2)\textquotedblright\ by \textquotedblleft% (3.1)\textquotedblright. \item \textbf{page 3:} \textquotedblleft by reversing all inequalities\textquotedblright\ $\rightarrow$ \textquotedblleft by reversing the inequalities (3.1) and (3.2)\textquotedblright\ (not the inequalities $\mu_{i}\lambda_{\pi(i)}-\pi(i)+i$, then we say that there exist no $\pi$-arrays. The \textit{weight} of a $\pi$-array means the product of $x_{a}$ for $a$ ranging over all entries of this array. We have% $s_{\lambda/\mu}^{R}=\sum_{\pi\in S_{k}}(-1)^{\pi}\cdot\left( \text{the sum of the weights of all }\pi\text{-arrays}\right)$ (this follows from the definition of $s_{\lambda/\mu}^{R}$ by writing the determinant as a sum over permutations). The involution $\varepsilon$ cancels unwanted terms in this formula (i.e., those which do not correspond to $\pi$ being the identity permutation and the $\pi$-array being an $R$-tableau) because it maps any $\pi$-array to a $\pi\circ\left( i,i+1\right)$-array, where $i$ is the row of the earliest failure. \item \textbf{page 13:} You write: \textquotedblleft Theorem 12 could easily be generalized to include part restrictions on the rows\textquotedblright. I suspect you mean Theorem 11, not Theorem 12, here. \item \textbf{page 17, proof of Lemma 18:} In the first displayed equation of this proof, replace \textquotedblleft$\left( a_{i}\right) _{j}%$\textquotedblright\ by \textquotedblleft$\left( \alpha_{i}\right) _{j}%$\textquotedblright. \item \textbf{page 18, proof of Lemma 19:} You write \textquotedblleft and the result follows from 18\textquotedblright. Probably you mean \textquotedblleft and the result follows from Lemma 18.\textquotedblright. \end{itemize} \end{document}