\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=Sunday, July 21, 2019 22:27:01} %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}} \newenvironment{proof2}[] {\begin{proof}[#1]}{\end{proof}} \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 Why should the LR rule be true?''} \ohead{\today} \begin{document} \begin{center} \textbf{Why should the Littlewood--Richardson rule be true?} \textit{Roger Howe and Soo Teck Lee} \url{https://doi.org/10.1090/S0273-0979-2011-01358-1} \texttt{howe lee - why LR.pdf} Bulletin of the American Mathematical Society \textbf{49} (2012), pp. 187--236 \textbf{Errata and addenda by Darij Grinberg} \bigskip \end{center} I will refer to the results appearing in the article \textquotedblleft% \textbf{Why should the Littlewood--Richardson rule be true?}\textquotedblright% \ by the numbers under which they appear in this article (specifically, in its published version). \textbf{The page numbers are relative to the article} (i.e., \textquotedblleft page 5\textquotedblright\ means \textquotedblleft the 5-th page of the article\textquotedblright, not \textquotedblleft the 5-th page of the issue\textquotedblright). \setcounter{section}{9} \section{Errata} My familiarity with this paper is fleeting. Thus, I will not be surprised if some of the following corrections are actually wrong; and even if not, I am fairly sure they are far from complete. \begin{itemize} \item \textbf{Page 5:} Replace \textquotedblleft any collection $J$% \textquotedblright\ by \textquotedblleft any strictly increasing sequence $J$\textquotedblright. \item \textbf{Page 5:} Replace \textquotedblleft to be the span the basis vectors\textquotedblright\ by \textquotedblleft to be the span of the basis vectors\textquotedblright. \item \textbf{Page 6:} The words \textquotedblleft for $1\leq a\leq d$\textquotedblright\ which are directly after (2.4) should actually be inside the displayed equation (2.4). \item \textbf{Page 6:} Add a whitespace in \textquotedblleft$U_{i}% ^{\operatorname*{opp}}$is the span\textquotedblright. \item \textbf{Page 7:} \textquotedblleft A partition $\alpha$ is specified by a weakly decreasing sequence of non-negative integers,\textquotedblright% \ should be replaced by \textquotedblleft A partition $\alpha$ is specified by a weakly decreasing sequence of non-negative integers such that all but finitely many of its entries are $0$.\textquotedblright. \item \textbf{Page 7:} In (2.14), replace \textquotedblleft$n-d-j_{d}% -d)$\textquotedblright\ by \textquotedblleft$n-d-\left( j_{d}-d\right)$\textquotedblright. \item \textbf{Page 10:} You write: \textquotedblleft It is not hard to argue that we can find an orthonormal basis $\left\{ \overrightarrow{\mathbf{y}% }_{b}\right\}$ for $V$, such that $\overrightarrow{\mathbf{y}}_{b}$ belongs to $U_{A,j_{b}}$\textquotedblright. It might be helpful to point out that this follows from Gram-Schmidt orthonormalization. \item \textbf{Page 10:} You write: \textquotedblleft if $J_{V}% ^{\operatorname*{opp}}=J^{\operatorname*{opp}}$ is the jump sequence of $V$ with respect to the opposite flag $\mathcal{F}_{A}^{\operatorname*{opp}}$ defined by the spaces $U_{A,n-j}^{\perp}$\textquotedblright. This notation conflicts with the definition of $J^{\operatorname*{opp}}$ on page 6, unless you mean to say that the $J^{\operatorname*{opp}}$ defined on page 6 actually \textbf{is} the jump sequence of $V$ with respect to the opposite flag $\mathcal{F}_{A}^{\operatorname*{opp}}$ -- but this, I believe, is false. Due to this confusion, I do not understand how you get $\operatorname*{tr}% \left( P_{V}A\right) \leq\sum\limits_{c=1}^{d}\lambda_{n-j_{c}% ^{\operatorname*{opp}}+1}\left( A\right)$ and prove Theorem 2.1. I also think there are further typos in these arguments: for example, I believe \textquotedblleft the intersection of Schubert varieties $\mathcal{S}% _{\mathcal{F}_{\left( A+B\right) ,K}}\cap\mathcal{S}_{\mathcal{F}% _{A,I}^{\operatorname*{opp}}}\cap\mathcal{S}_{\mathcal{F}_{B,J}% ^{\operatorname*{opp}}}$\textquotedblright\ should be \textquotedblleft the intersection of Schubert varieties $\Omega_{\mathcal{F}_{A+B,K}}\cap \Omega_{\mathcal{F}_{A}^{\operatorname*{opp}},I}\cap\Omega_{\mathcal{F}% _{B}^{\operatorname*{opp}},J}$\textquotedblright\ on page 10, and I am also wondering if the \textquotedblleft$c_{\alpha_{J},\alpha_{K}}^{\alpha _{I}^{\operatorname*{opp}}}$\textquotedblright\ in Theorem 2.1 shouldn't rather be something like \textquotedblleft$c_{\alpha_{I^{\operatorname*{opp}}% },\alpha_{J^{\operatorname*{opp}}}}^{\alpha_{K}}$\textquotedblright. \item \textbf{Page 11, Theorem 2.1:} Replace \textquotedblleft the the\textquotedblright\ by \textquotedblleft the\textquotedblright. \item \textbf{Page 12, \S 3:} Replace \textquotedblleft for any vector $\mathbf{v}$ in $V$\textquotedblright\ by \textquotedblleft for any vector $v$ in $V$\textquotedblright. \item \textbf{Page 12, \S 3:} On the same line, replace \textquotedblleft the function on $G$\textquotedblright\ by \textquotedblleft the function on $\operatorname*{GL}\nolimits_{n}$\textquotedblright. (Or maybe define $G=\operatorname*{GL}\nolimits_{n}$, if you call it $G$ again later.) \item \textbf{Page 13:} \textquotedblleft repsentations\textquotedblright% \ $\rightarrow$ \textquotedblleft representations\textquotedblright. \item \textbf{Page 15:} In condition (ii'), replace \textquotedblleft for $j,a\geq1$\textquotedblright\ by \textquotedblleft for $j\geq1$ and $a\geq 0$\textquotedblright. \item \textbf{Page 16:} Replace \textquotedblleft skew-row\textquotedblright% \ by \textquotedblleft skew row\textquotedblright\ (twice). \item \textbf{Page 17:} In \textquotedblleft by a nested sequence $D=D_{0}\subset D_{1}^{\prime}\subset D_{2}^{\prime}\subset\cdots\subset D_{r}^{\prime}$ of Young diagrams\textquotedblright, replace \textquotedblleft% $D_{0}$\textquotedblright\ by \textquotedblleft$D_{0}^{\prime}$% \textquotedblright. \item \textbf{Page 19:} You write: \textquotedblleft Suppose that the row lengths $a_{j}$ are weakly decreasing\textquotedblright. I find it unmotivated that you refer to the $a_{j}$ as \textquotedblleft row lengths\textquotedblright\ here, since so far they are just nonnegative integers, and I don't think you have declared your intention to consider them as row lengths of a Young diagram. \item \textbf{Page 20:} \textquotedblleft nubmers\textquotedblright% \ $\rightarrow$ \textquotedblleft numbers\textquotedblright. \item \textbf{Page 22:} Replace \textquotedblleft among all the possible tableau\textquotedblright\ by \textquotedblleft among all the possible tableaux\textquotedblright. \item \textbf{Page 22, Lemma 6.1:} It would be good to add the sentence \textquotedblleft Let $a$ and $b$ be nonnegative integers satisfying $a\geq b$\textquotedblright\ at the beginning of this lemma. This would remind the reader of the standing assumption that $a\geq b$ (which was briefly mentioned on page 21, but is easily overlooked or understood to only apply to page 21). (Actually, the slightly weaker assumption $a\geq b-1$ is enough for your proof to work.) \item \textbf{Page 23, proof of Lemma 6.1:} Replace \textquotedblleft We look at the first point $p_{o}=% %TCIMACRO{\QDATOPD{[}{]}{n}{m}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{n}{m}% %EndExpansion$\textquotedblright\ by \textquotedblleft We look at the first point $p_{o}=% %TCIMACRO{\QDATOPD{[}{]}{m}{n}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{m}{n}% %EndExpansion$\textquotedblright. \item \textbf{Page 23, proof of Lemma 6.1:} Replace \textquotedblleft and $n$ is the smallest\textquotedblright\ by \textquotedblleft and $m$ is the smallest\textquotedblright. \item \textbf{Page 23, proof of Lemma 6.1:} Replace \textquotedblleft so that if $p_{o}$ is not the origin, then $h\left( p_{o}\right) >0$; that is, $p_{o}$ lies strictly above the diagonal\textquotedblright\ by \textquotedblleft but $h\left( p_{o}\right) >0$ (since $\mathcal{P}$ rises above the main diagonal); thus, $p_{o}$ lies strictly above the main diagonal. In particular, $p_{o}$ is not the origin.\textquotedblright. \item \textbf{Page 23, proof of Lemma 6.1:} Replace \textquotedblleft$% %TCIMACRO{\QDATOPD{[}{]}{m-1}{n}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{m-1}{n}% %EndExpansion$\textquotedblright\ by \textquotedblleft$% %TCIMACRO{\QDATOPD{[}{]}{m}{n-1}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{m}{n-1}% %EndExpansion$\textquotedblright\ (three times in the proof). \item \textbf{Page 23, proof of Lemma 6.1:} Remove the words \textquotedblleft the next move of $\mathcal{P}$ must be to the right, that is, the next point after $p_{o}$ on $\mathcal{P}$ must be $% %TCIMACRO{\QDATOPD{[}{]}{m+1}{n}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{m+1}{n}% %EndExpansion$\textquotedblright. You never use this observation. \item \textbf{Page 23, proof of Lemma 6.1:} Replace \textquotedblleft by shifting by $% %TCIMACRO{\QDATOPD{[}{]}{-1}{1}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{-1}{1}% %EndExpansion$\textquotedblright\ by \textquotedblleft by shifting by $% %TCIMACRO{\QDATOPD{[}{]}{1}{-1}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{1}{-1}% %EndExpansion$\textquotedblright. \item \textbf{Page 24, proof of Lemma 6.1:} Replace \textquotedblleft$% %TCIMACRO{\QDATOPD{[}{]}{m^{\prime}}{n^{\prime}}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{m^{\prime}}{n^{\prime}}% %EndExpansion$on\textquotedblright\ by \textquotedblleft$% %TCIMACRO{\QDATOPD{[}{]}{m^{\prime}}{n^{\prime}}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{m^{\prime}}{n^{\prime}}% %EndExpansion$ on\textquotedblright. \item \textbf{Page 24, proof of Lemma 6.1:} There is a subtlety here that should (in my opinion) be made explicit. You speak of \textquotedblleft the path $\mathcal{P}$ constructed in the previous paragraph from $\mathcal{P}% ^{\prime}$\textquotedblright. This construction of $\mathcal{P}$ from $\mathcal{P}^{\prime}$ rests on one assumption: the assumption that the last point on the highest diagonal reached by $\mathcal{P}^{\prime}$ is not the endpoint of the path $\mathcal{P}^{\prime}$.\ \ \ \ \footnote{Indeed, you use this assumption (because you speak of \textquotedblleft The move from this point\textquotedblright, and this move only exists if this point is not the endpoint).} This assumption, of course, is obviously satisfied when your path $\mathcal{P}^{\prime}$ results from an increasing path $\mathcal{P}$ by the algorithm you explained on page 23, but it is not completely obvious why it holds when the path $\mathcal{P}^{\prime}$ is just some arbitrary increasing path from $% %TCIMACRO{\QDATOPD{[}{]}{0}{0}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{0}{0}% %EndExpansion$ to $% %TCIMACRO{\QDATOPD{[}{]}{a+1}{b-1}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{a+1}{b-1}% %EndExpansion$. So let me prove it in the latter case. Let $\mathcal{P}^{\prime}$ be an increasing path from $% %TCIMACRO{\QDATOPD{[}{]}{0}{0}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{0}{0}% %EndExpansion$ to $% %TCIMACRO{\QDATOPD{[}{]}{a+1}{b-1}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{a+1}{b-1}% %EndExpansion$. Then, $a+1>a\geq b-1$, so that the point $% %TCIMACRO{\QDATOPD{[}{]}{a+1}{b-1}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{a+1}{b-1}% %EndExpansion$ lies (strictly) below the main diagonal. Therefore, the origin lies on a higher diagonal than the point $% %TCIMACRO{\QDATOPD{[}{]}{a+1}{b-1}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{a+1}{b-1}% %EndExpansion$. Hence, the path $\mathcal{P}^{\prime}$ reaches a point on a higher diagonal than $% %TCIMACRO{\QDATOPD{[}{]}{a+1}{b-1}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{a+1}{b-1}% %EndExpansion$ (namely, the origin). Therefore, the last point on the highest diagonal reached by $\mathcal{P}^{\prime}$ is not $% %TCIMACRO{\QDATOPD{[}{]}{a+1}{b-1}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{a+1}{b-1}% %EndExpansion$. In other words, the last point on the highest diagonal reached by $\mathcal{P}^{\prime}$ is not the endpoint of the path $\mathcal{P}^{\prime}$ (since the endpoint of the path $\mathcal{P}^{\prime}$ is $% %TCIMACRO{\QDATOPD{[}{]}{a+1}{b-1}}% %BeginExpansion \genfrac{[}{]}{0pt}{0}{a+1}{b-1}% %EndExpansion$). This finishes the proof. \item \textbf{Page 24, proof of Lemma 6.1:} Replace \textquotedblleft% $P\rightarrow P^{\prime}$\textquotedblright\ by \textquotedblleft% $\mathcal{P}\mapsto\mathcal{P}^{\prime}$\textquotedblright. \item \textbf{Page 24:} Replace \textquotedblleft$\rho_{n}^{D}\times S^{2}\otimes S^{2}$\textquotedblright\ by \textquotedblleft$\rho_{n}% ^{D}\otimes S^{2}\otimes S^{2}$\textquotedblright. \item \textbf{Page 30:} Replace \textquotedblleft we will show that how\textquotedblright\ by either \textquotedblleft we will show that\textquotedblright\ or \textquotedblleft we will show how\textquotedblright. \item \textbf{Page 35:} Replace \textquotedblleft that a tableaux\textquotedblright\ by \textquotedblleft that a tableau\textquotedblright. \item \textbf{Page 39:} \textquotedblleft in a the cone\textquotedblright% \ $\rightarrow$ \textquotedblleft in the cone\textquotedblright. \item \textbf{Page 39:} \textquotedblleft to the Hibi ring $\mathbb{C}\left( \mathbb{Z}_{\geq}^{+}\right) \left( \operatorname*{GT}\nolimits_{\left( n,k,\ell\right) }\right)$\textquotedblright\ $\rightarrow$ \textquotedblleft to the Hibi ring $\mathbb{C}\left( \mathbb{Z}_{\geq}% ^{+}\left( \operatorname*{GT}\nolimits_{\left( n,k,\ell\right) }\right) \right)$\textquotedblright. \item \textbf{Page 40:} Replace \textquotedblleft Knutson-Tau\textquotedblright\ by \textquotedblleft Knutson-Tao\textquotedblright. \item \textbf{Page 48, reference [Hum]:} Replace \textquotedblleft Humphrey\textquotedblright\ by \textquotedblleft Humphreys\textquotedblright. \item \textbf{Page 48, reference [Rei]:} The title of this reference should be \textquotedblleft Signed poset\textquotedblright. The \textquotedblleft Victor\textquotedblright\ is just the first name of the author. \end{itemize} \end{document}