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\title[Corrigendum to ``A colourful path to matrix-tree theorems''] {Corrigendum to\texorpdfstring{\\}{} ``A colourful path to matrix-tree theorems''}

\author{\firstname{Adrien} \lastname{Kassel}}
\address{UMPA, CNRS, ENS de Lyon, 46 all\'{e}e d'Italie, 69364 Lyon Cedex 07 (France)}
\email{adrien.kassel@ens-lyon.fr}

\author{\firstname{Thierry}  \lastname{L\'{e}vy}}
\address{LPSM, Sorbonne Universit\'{e}, 4 place Jussieu, 75005 Paris (France)}
\email{thierry.levy@sorbonne-universite.fr}

\keywords{matrix-tree theorem, determinant, Q-determinant}
  
\subjclass[2010]{05C30, 05C22, 15A15}

\begin{document}

\begin{abstract}
Some statements in the original publication relied on an implicit assumption. This corrigendum provides the necessary modifications when that assumption does not hold.
\end{abstract}

\maketitle

\theoremstyle{plain} 
\newtheorem{theo2}{Theorem}
\newtheorem{coro2}[theo2]{Corollary}

In \cite[Eq. (1)]{KL2}, we defined a notion of $\tau$-determinant for a general tracial map $\tau$. However, in our combinatorial expansion results, we implicitly made the assumption that $\tau(1)=1$. In this short note, we reformulate the results of \cite{KL2}, using the same notation, in the case where this equality is not assumed. 

To be clear, the results of \cite{KL2} are correct as we originally stated them under the assumption that $\tau(1)=1$. Without this assumption, and using the same proofs taking into account the value of $\tau(1)$ in the computations (which adds a weight $\tau(1)$ per ``black arrow'' in the graphical expansions), one easily finds the following modified statements.

\begin{theo2}[{Corrects \cite[Theorem 3.1]{KL2}}]
In the ring $S=K[a_{ij}\colon (i,j)\in\rE]$, 
\begin{equation*}
{\det}_{\tau} \Delta_{[m]}=\tau(1)^m \sum_{\rF\in\scF_m} a_{\rF} \prod_{c\in\mathscr{C}(\rF)} \left(1-\tau(1)^{-\ell(c)}\tau(h_c)\right)\,.
\end{equation*}
\end{theo2}

\begin{coro2}[{Corrects \cite[Corollary 3.2]{KL2}}]
In the quotient $S/\big(a_{ij}-a_{ji}\colon (i,j)\in\rE\big)$, 
\begin{align*}
&{\det}_{\tau} \Delta_{[m]}=\tau(1)^m \sum_{[\rF]\in\mathscr{F}_m} a_{\rF} \prod_{\substack{c\in\mathscr{C}(\rF)\\ \ell(c)=2}} \left(1-\tau(1)^{-\ell(c)}\tau(h_c)\right)\\ 
& \hspace{6cm}\prod_{\substack{c\in\mathscr{C}(\rF)\\ \ell(c)\ge 3}} \left(2-\tau(1)^{-\ell(c)}\big(\tau(h_c)+\tau(h_{c^{-1}})\big)\right)\,.
\end{align*}
\end{coro2}

\begin{theo2}[{Corrects \cite[Theorem 5.1]{KL2}}]
In the ring $S=K[a_{ij}\colon (i,j)\in \E]$, 
\begin{align*}
&{\det}_{\tau}  \Delta_{[Nm]}=\tau(1)^{Nm}\sum_{\rF\in \HF_{m,N}} a_{\rF} \prod_{c\in \Hcyc(\rF)} \big(1-\tau(1)^{-\ell(c)}\tau(\hb_{c})\big) \\
&\hspace{8cm} \prod_{c\in \mathscr{S}(\rF)}\big(-\tau(1)^{-\ell(c)}\tau(\hb_{c})\big)\,.
\end{align*}
\end{theo2}

\begin{coro2}[{Corrects \cite[Corollary 5.2]{KL2}}]
In the quotient $S/(a_{ij}-a_{ji}\colon (i,j)\in \E)$,
\begin{align*}
{\det}_{\tau}  \Delta_{[Nm]}=& \tau(1)^{Nm}\sum_{[\rF]\in \mathscr{H}\!\mathscr{F}_{m,N}} a_{\rF} \prod_{\substack{c\in \Hcyc(\rF)\\ \ell(c)=2}} \big(1- \tau(1)^{-\ell(c)}\tau(h_{c})\big) \\
& \hspace{1cm} \prod_{\substack{c\in \Hcyc(\rF)\\ \ell(c)\geq 3}} \Big(2- \tau(1)^{-\ell(c)}\big(\tau(h_{c})+\tau(h_{c^{-1}})\big)\Big)\\
&\hspace{1cm}\prod_{\substack{c\in \mathscr{S}(\rF)\\ \ell(c)=2}} (-\tau(1))^{-\ell(c)}\tau(h_{c}) \prod_{\substack{c\in \mathscr{S}(\rF)\\ \ell(c)\geq 3}} (-\tau(1))^{-\ell(c)}\big(\tau(h_{c})+ \tau(h_{c^{-1}})\big)
\,.
\end{align*}
\end{coro2}

\begin{theo2}[{Corrects \cite[Theorem 6.1]{KL2}}]
In the ring $S=K[a_{ij}\colon (i,j)\in \rE]$,
\begin{equation*}
    {\det}_{\tau} \Delta_{[m]}=\left[\frac{\tau(1)}{d}\right]^m \sum_{\rF\in\scF_m} a_{\rF} \prod_{c\in\mathscr{C}(\rF)} \left(1-\left[\frac{d}{\tau(1)}\right]^{\ell(c)}\varepsilon_c \tau(h_c)\right)\,.
\end{equation*}
\end{theo2}

\begin{theo2}[{Corrects \cite[Theorem 7.1]{KL2}}]
In the ring $S=K[a_{ij}\colon (i,j)\in\rE]$, 
\begin{equation*}
{\det}_{\tau} \Delta_{[Nm]}=\bigg[\frac{\tau(1)}{N}\bigg]^{mN} \sum_{\rF\in\scF_{m,N}} a_{\rF} \prod_{c\in\mathscr{C}(\rF)} \left(1-\bigg[\frac{N}{\tau(1)}\bigg]^{\ell(c)}\tau(\hb_c)\right)\,.
\end{equation*}
\end{theo2}
\medskip

In particular, when $\tau(1)=N$ (which is the case when taking $H=M_N(\C)$, $K=\C$, and $\tau:H\to K$ given by the trace $\tau(\cdot)=\Tr(\cdot)$) the last equation simplifies nicely to give the following corollary which did not appear in \cite{KL2}.

\begin{coro2}[{Not stated in \cite{KL2}}]
Assume that $\tau(1)=N$. Then the following equality holds in the ring $S=K[a_{ij}: (i,j)\in\rE]$:
\begin{equation*}
{\det}_{\tau} \Delta_{[Nm]}= \sum_{\rF\in\scF_{m,N}} a_{\rF} \prod_{c\in\mathscr{C}(\rF)} \left(1-\tau(\hb_c)\right)\,.
\end{equation*}
\end{coro2}
\medskip

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