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\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 [4].
\end{itemize}
\end{document}