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\ihead{Errata to ``noncommutative geometry@n, volume 1''}
\ohead{\today}
\begin{document}

\begin{center}
\textbf{noncommutative geometry@n, volume 1 : the tools}

\textit{Lieven Le Bruyn}

\url{http://win.ua.ac.be/~lebruyn/LeBruyn2005d.pdf}

version of 19 March 2006

\textbf{Errata and addenda by Darij Grinberg}

\bigskip
\end{center}

The following is a haphazard list of errors I found in \textquotedblleft
noncommutative geometry@n, volume 1 : the tools\textquotedblright\ by Lieven
Le Bruyn.

All page numbers given below are to be understood as page numbers in the
printed book (thus, page 1 is the first page of the Introduction, and not the
first page of the PDF file).

\setcounter{section}{10}

\section{Errata}

\begin{itemize}
\item \textbf{Page 7:} Add a whitespace before \textquotedblleft
This\textquotedblright\ in \textquotedblleft relations.This\textquotedblright.

\item \textbf{Page 11:} In the formula \textquotedblleft$\sigma_{l}\left(
\lambda_{1},\ldots,\lambda_{l}\right)  =\sum_{i_{1}<i_{2}<\ldots<i_{l}}%
\lambda_{i_{1}}\lambda_{i_{2}}\ldots\lambda_{i_{l}}$\textquotedblright,
replace \textquotedblleft$\lambda_{l}$\textquotedblright\ by \textquotedblleft%
$\lambda_{n}$\textquotedblright.

Also, it makes sense to define $\sigma_{l}\left(  A\right)  :=\sigma
_{l}\left(  \lambda_{1},\ldots,\lambda_{n}\right)  $; this notation will be
used just a few lines below.

\item \textbf{Page 18, Theorem 1.3:} Replace \textquotedblleft is a
regular\textquotedblright\ by \textquotedblleft be a regular\textquotedblright.

\item \textbf{Page 18, proof of Theorem 1.3:} Replace \textquotedblleft
functions on $\mathbb{C}^{m}$\textquotedblright\ by \textquotedblleft
functions on $\mathbb{C}^{n}$\textquotedblright.

\item \textbf{Page 18, proof of Theorem 1.3:} Replace \textquotedblleft%
$f^{\prime}\in\mathbb{C}\left[  \sigma_{1}\left(  X\right)  ,\ldots,\sigma
_{n}\left(  X\right)  \right]  $\textquotedblright\ by \textquotedblleft%
$f^{\prime}\in\mathbb{C}\left[  \sigma_{1},\ldots,\sigma_{n}\right]  $, and
thus $f^{\prime}\circ\pi\in\mathbb{C}\left[  \sigma_{1}\left(  X\right)
,\ldots,\sigma_{n}\left(  X\right)  \right]  $\textquotedblright.

\item \textbf{Page 18, proof of Theorem 1.3:} Replace \textquotedblleft
Moreover, $f$ and $f^{\prime}$\textquotedblright\ by \textquotedblleft
Moreover, $f$ and $f^{\prime}\circ\pi$\textquotedblright.

\item \textbf{Page 18:} Replace \textquotedblleft$A\in A$\textquotedblright%
\ by \textquotedblleft$A\in M_{n}$\textquotedblright.

\item \textbf{Page 19:} \textquotedblleft defined by mapping $\lambda
_{j+1},\ldots,\lambda_{j}$ to zero\textquotedblright\ $\rightarrow$
\textquotedblleft defined by mapping $\lambda_{j+1},\ldots,\lambda_{n}$ to
zero\textquotedblright.

\item \textbf{Page 19:} In the last displayed formula on this page, replace
\textquotedblleft$n-i$\textquotedblright\ by \textquotedblleft$j-i$%
\textquotedblright\ (in \textquotedblleft$\sum_{i=1}^{j}\left(  -1\right)
^{i}\phi\left(  \sigma_{i}\right)  \phi\left(  s_{n-i}\right)  $%
\textquotedblright).

\item \textbf{Page 20:} \textquotedblleft with diagonal entries $\left(
\lambda_{1},\ldots,\lambda_{1}\right)  $\textquotedblright\ $\rightarrow$
\textquotedblleft with diagonal entries $\left(  \lambda_{1},\ldots
,\lambda_{n}\right)  $\textquotedblright.

\item \textbf{Page 20, Theorem 1.4:} In the upper right corner of the matrix,
replace \textquotedblleft$x_{nn}$\textquotedblright\ by \textquotedblleft%
$x_{1n}$\textquotedblright.

\item \textbf{Page 26:} The displayed formula \textquotedblleft$\sigma.\left(
v_{1}\otimes\cdots\otimes v_{m}\right)  =v_{\sigma\left(  1\right)  }%
\otimes\cdots\otimes v_{\sigma\left(  m\right)  }$\textquotedblright\ does
\textbf{not} define a left action of $S_{m}$ on $V_{n}^{\otimes m}$. It should
be replaced either by the formula \textquotedblleft$\left(  v_{1}\otimes
\cdots\otimes v_{m}\right)  .\sigma=v_{\sigma\left(  1\right)  }\otimes
\cdots\otimes v_{\sigma\left(  m\right)  }$\textquotedblright\ defining a
\textbf{right} action of $S_{m}$ on $V_{n}^{\otimes m}$, or by the formula
\textquotedblleft$\sigma.\left(  v_{1}\otimes\cdots\otimes v_{m}\right)
=v_{\sigma^{-1}\left(  1\right)  }\otimes\cdots\otimes v_{\sigma^{-1}\left(
m\right)  }$\textquotedblright\ defining a left action of $S_{m}$ on
$V_{n}^{\otimes m}$. (In either case, I'm afraid it might be necessary to make
some more replacements further on.)

\item \textbf{Page 27, proof of Theorem 1.5:} In the proof of part (2),
replace \textquotedblleft$End\left(  V_{n}^{\otimes m}\right)  \simeq M_{nm}%
$\textquotedblright\ by \textquotedblleft$End\left(  V_{n}^{\otimes m}\right)
\simeq M_{n^{m}}$\textquotedblright.

\item \textbf{Page 27, proof of Theorem 1.5:} In the proof of part (2),
replace \textquotedblleft$End_{S_{m}}\left(  V_{m}^{\otimes m}\right)
$\textquotedblright\ by \textquotedblleft$End_{S_{m}}\left(  V_{n}^{\otimes
m}\right)  $\textquotedblright.

\item \textbf{Page 27:} In the displayed formula preceding Proposition 1.1,
you write: \textquotedblleft$\prod_{i}f_{i}\left(  v_{\sigma^{(}i)}\right)
$\textquotedblright. This should be \textquotedblleft$\prod_{i}f_{i}\left(
v_{\sigma\left(  i\right)  }\right)  $\textquotedblright.

\item \textbf{Page 27, Proposition 1.1:} Add a tensor sign between
\textquotedblleft$\cdots$\textquotedblright\ and \textquotedblleft$f_{m}%
$\textquotedblright\ in \textquotedblleft$f_{1}\otimes\cdots f_{m}\otimes
v_{1}\otimes\cdots\otimes v_{m}$\textquotedblright.

The same mistake appears in the displayed formula preceding Proposition 1.1.
It also appears one line above this formula.

Furthermore, the displayed formula preceding Proposition 1.1 has another
similar mistake: a tensor sign is missing between \textquotedblleft$\cdots
$\textquotedblright\ and \textquotedblleft$v_{\sigma\left(  m\right)  }%
$\textquotedblright\ in \textquotedblleft$v_{\sigma\left(  1\right)  }%
\otimes\cdots v_{\sigma\left(  m\right)  }$\textquotedblright.

\item \textbf{Page 30:} In the first displayed formula on this page, replace
\textquotedblleft$t_{d_{1}}$\textquotedblright\ by \textquotedblleft$t_{d_{1}%
}\left(  1\right)  $\textquotedblright.

\item \textbf{Page 30:} In the long displayed formula that follows the words
\textquotedblleft then we get an expression\textquotedblright, replace
\textquotedblleft$t_{d_{1}}^{s_{d_{1}}\left(  1\right)  }$\textquotedblright%
\ by \textquotedblleft$t_{d_{1}}\left(  1\right)  ^{s_{d_{1}}\left(  1\right)
}$\textquotedblright.

\item \textbf{Page 31:} Replace \textquotedblleft to obtain an $n\times n$
matrix with coefficients in $M_{n}\left(  \mathbb{C}\left[  M_{n}^{m}\right]
\right)  $\textquotedblright\ by \textquotedblleft to obtain an $n\times n$
matrix in $M_{n}\left(  \mathbb{C}\left[  M_{n}^{m}\right]  \right)
$\textquotedblright\ (or by \textquotedblleft to obtain an $n\times n$ matrix
with entries in $\mathbb{C}\left[  M_{n}^{m}\right]  $\textquotedblright).

\item \textbf{Page 32:} Replace \textquotedblleft$X_{i}:M_{n}^{m}=M_{n}%
\oplus\ldots\oplus M_{n}^{m}\rightarrow M_{n}$\textquotedblright\ by
\textquotedblleft$X_{i}:M_{n}^{m}=M_{n}\oplus\ldots\oplus M_{n}\rightarrow
M_{n}$\textquotedblright\ (there was one \textquotedblleft$m$%
\textquotedblright\ exponent too much).

\item \textbf{Page 33, proof of Proposition 1.3:} You write: \textquotedblleft
By theorem 1.6 we can therefore write%
\begin{equation}
tr\left(  fX_{m+1}\right)  =\sum\underbrace{g_{i_{1}\ldots i_{l}}}%
_{\in\mathbb{N}_{n}^{m}}tr\left(  X_{i_{1}}\ldots X_{i_{l}}X_{m+1}\right)
\label{p33.proof-of-prop-1-3.1}%
\end{equation}
\textquotedblright. I would suggest being a bit more precise here: First,
replace the \textquotedblleft$\sum$\textquotedblright\ sign by
\textquotedblleft$\sum_{i_{1},\ldots,i_{l}\in\left\{  1,2,\ldots,m\right\}  }%
$\textquotedblright. Secondly, it is worth explaining why $tr\left(
fX_{m+1}\right)  $ can be represented in the form
(\ref{p33.proof-of-prop-1-3.1}): Namely, theorem 1.6 shows that $tr\left(
fX_{m+1}\right)  $ can be represented as a polynomial in the invariants
$tr\left(  X_{i_{1}}\ldots X_{i_{l}}\right)  $ (where $i_{1},\ldots,i_{l}%
\in\left\{  1,2,\ldots,m+1\right\}  $). In other words, $tr\left(
fX_{m+1}\right)  $ is a linear combination of products of the form $tr\left(
X_{a_{1}}\ldots X_{a_{l}}\right)  tr\left(  X_{b_{1}}\ldots X_{b_{m}}\right)
\ldots tr\left(  X_{h_{1}}\ldots X_{h_{s}}\right)  $. Furthermore, in each
such product, the matrix $X_{m+1}$ occurs exactly once (since $tr\left(
fX_{m+1}\right)  $ is linear in $X_{m+1}$); thus, we can push the unique
factor in which $X_{m+1}$ occurs to the end of the product and also push the
$X_{m+1}$ to the end of this factor. As a result, the product rewrites in the
form $\underbrace{g_{i_{1}\ldots i_{l}}}_{\in\mathbb{N}_{n}^{m}}tr\left(
X_{i_{1}}\ldots X_{i_{l}}X_{m+1}\right)  $ for some $i_{1},\ldots,i_{l}%
\in\left\{  1,2,\ldots,m\right\}  $. Altogether, we obtain a representation in
the form (\ref{p33.proof-of-prop-1-3.1}).

\item \textbf{Page 34, Proposition 1.4:} Replace \textquotedblleft$l\leq
2^{n}-1$\textquotedblright\ by \textquotedblleft$l\leq2^{n}-2$%
\textquotedblright. (I am not saying that \textquotedblleft$l\leq2^{n}%
-1$\textquotedblright\ is false; but your proof yields the better bound
\textquotedblleft$l\leq2^{n}-2$\textquotedblright.)

\item \textbf{Page 34, proof of Proposition 1.4:} Replace \textquotedblleft%
$R=\mathbb{T}_{+}/\mathbb{N}_{+}.\mathbb{T}_{+}$\textquotedblright\ by
\textquotedblleft$R=\mathbb{T}_{+}/\mathbb{N}_{+}.\mathbb{T}$%
\textquotedblright. (With your definition \textquotedblleft$R=\mathbb{T}%
_{+}/\mathbb{N}_{+}.\mathbb{T}_{+}$\textquotedblright, your claim that
\textquotedblleft for any $x\in R$ we have that $x^{n}=0$\textquotedblright%
\ would be unfounded, because the last term $c_{n}$ on the left hand side of
the equation \textquotedblleft$t^{n}+c_{1}t^{n-1}+\ldots+c_{n}=0$%
\textquotedblright\ does not lie in $\mathbb{N}_{+}.\mathbb{T}_{+}$.)

\item \textbf{Page 34, proof of Proposition 1.4:} Replace \textquotedblleft
degree at most $2^{n}-1$\textquotedblright\ by \textquotedblleft degree at
most $2^{n}-2$\textquotedblright.

\item \textbf{Page 35, Theorem 1.8:} Replace \textquotedblleft of length
$l\leq2^{n}$\textquotedblright\ by \textquotedblleft of length $l\leq2^{n}%
-1$\textquotedblright.

\item \textbf{Page 35, proof of Theorem 1.8:} Replace \textquotedblleft of
degree at most $2^{n}-1$\textquotedblright\ by \textquotedblleft of degree at
most $2^{n}-2$\textquotedblright.

\item \textbf{Page 35, proof of Theorem 1.8:} Replace \textquotedblleft of
degree at most $2^{n}$.\textquotedblright\ by \textquotedblleft of degree at
most $2^{n}-1$.\textquotedblright.

\item \textbf{Page 35, proof of Theorem 1.8:} Replace \textquotedblleft by all
$tr\left(  S^{\prime}\right)  $\textquotedblright\ by \textquotedblleft by
$tr\left(  \mathbb{S}^{\prime}\right)  $\textquotedblright.

\item \textbf{Page 35, proof of Theorem 1.8:} Replace \textquotedblleft%
$\mathbb{N}_{+}=\mathbb{N}_{n}^{m}tr\left(  \mathbb{T}_{+}\right)
$\textquotedblright\ by \textquotedblleft$\mathbb{N}_{+}=\mathbb{N}_{n}%
^{m}tr\left(  \mathbb{T}_{+}^{\prime}\right)  $\textquotedblright.

\item \textbf{Page 36, Example 1.4:} Replace \textquotedblleft$det\left(
X_{i}\right)  =\dfrac{1}{2}\left(  tr\left(  X_{i}\right)  ^{2}-tr\left(
X_{i}\right)  ^{2}\right)  $\textquotedblright\ by \textquotedblleft%
$det\left(  X_{i}\right)  =\dfrac{1}{2}\left(  \left(  tr\left(  X_{i}\right)
\right)  ^{2}-tr\left(  X_{i}^{2}\right)  \right)  $\textquotedblright.

\item \textbf{Page 37:} On the first line of this page, replace
\textquotedblleft tableaux\textquotedblright\ by \textquotedblleft
tableau\textquotedblright.

\item \textbf{Page 37:} Replace \textquotedblleft define different
subgroups\textquotedblright\ by \textquotedblleft sometimes define different
subgroups\textquotedblright. (It is easy to find two distinct tableaux of
shape $\left(  2,2\right)  $ which define the same $P_{\lambda}$ and the same
$Q_{\lambda}$.)

\item \textbf{Page 37:} In the definition of $c_{\lambda}$, replace
\textquotedblleft$c_{\lambda}=a_{\lambda}.b_{\sigma}$\textquotedblright\ by
\textquotedblleft$c_{\lambda}=a_{\lambda}.b_{\lambda}$\textquotedblright.

\item \textbf{Page 37, Theorem 1.9:} \textquotedblleft
representations\textquotedblright\ $\rightarrow$ \textquotedblleft
representation\textquotedblright.

\item \textbf{Page 37, proof of Theorem 1.9:} \textquotedblleft consider the
tableaux\textquotedblright\ $\rightarrow$ \textquotedblleft consider the
tableau\textquotedblright.

\item \textbf{Page 37, proof of Theorem 1.9:} \textquotedblleft there is an
element $p_{1}\in T_{\lambda}$\textquotedblright\ should be \textquotedblleft
there is an element $p_{1}\in P_{\lambda}$\textquotedblright.

\item \textbf{Page 37, proof of Theorem 1.9:} \textquotedblleft$q^{\prime}%
\in\sigma.Q_{\lambda},\sigma^{-1}$\textquotedblright\ should be
\textquotedblleft$q^{\prime}\in\sigma.Q_{\lambda}.\sigma^{-1}$%
\textquotedblright.

\item \textbf{Page 39:} Do you count $t\left(  1\right)  $ among the
generators of $\mathbb{N}^{\infty}$ or not? (I suspect you don't, but I am not
fully sure.)

Also, I think your map $t:\mathbb{T}^{\infty}\rightarrow\mathbb{N}^{\infty}$
is not surjective, in spite of you using the $\twoheadrightarrow$ arrow when
defining it. In fact, its image is $\mathbb{N}_{+}^{\infty}$.

\item \textbf{Page 41:} You again define an action of $S_{d}$ on $V^{\otimes
d}$ by \textquotedblleft$\sigma.\left(  v_{1}\otimes\cdots\otimes
v_{d}\right)  =v_{\sigma\left(  1\right)  }\otimes\cdots\otimes v_{\sigma
\left(  d\right)  }$\textquotedblright\ (as on page 26). Again, this does not
define a valid left action of $S_{d}$ on $V^{\otimes d}$.

\item \textbf{Page 41:} Replace \textquotedblleft endomorphism $\sum
a_{\sigma}\lambda_{\sigma}\in End\left(  V^{\otimes m}\right)  $%
\textquotedblright\ by \textquotedblleft endomorphism $\sum a_{\sigma}%
\lambda_{\sigma}\in End\left(  V^{\otimes d}\right)  $\textquotedblright.

\item \textbf{Page 42:} Replace \textquotedblleft Then, the subgroup
$Q_{\lambda}$ of $S_{d}$ which preserves each row of $\lambda$ (or
equivalently, each column of $\lambda^{\ast}$) is\textquotedblright\ by
\textquotedblleft Then, the subgroup $Q_{\lambda}$ of $S_{d}$ which preserves
each row of $\lambda^{\ast}$ (or equivalently, each column of $\lambda$)
is\textquotedblright.

\item \textbf{Page 42:} You say that \textquotedblleft we have
proved\textquotedblright\ Theorem 1.10. But is this so? I see that you have
proved the \textquotedblleft if\textquotedblright\ direction of Theorem 1.10
(since you have shown that any Young symmetrizer $c_{\lambda}$ corresponding
to a partition $\lambda$ with at least $n+1$ rows acts as zero on $V^{\otimes
d}$). But the \textquotedblleft only if\textquotedblright\ direction does not
seem to follow from what you have done.

Also, have you shown that the Young symmetrizers $c_{\lambda}$ relative to
partitions $\lambda$ with at least $n+1$ rows actually span an ideal of
$\mathbb{C}S_{d}$ ?

(The \textquotedblleft only if\textquotedblright\ direction can be derived
from Corollary 6.6 in William Fulton, Joe Harris, \textit{Representation
Theory: A First Course}, Springer 2004. However, this requires some nontrivial
representation theory. The claim that the Young symmetrizers $c_{\lambda}$
relative to partitions $\lambda$ with at least $n+1$ rows actually span an
ideal of $\mathbb{C}S_{d}$ can be derived from the \textquotedblleft
if\textquotedblright\ and \textquotedblleft only if\textquotedblright\ directions.)

\item \textbf{Page 43, Theorem 1.10:} Replace \textquotedblleft with a
least\textquotedblright\ by \textquotedblleft with at least\textquotedblright.
\end{itemize}


\end{document}