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\begin{document}

\newbox\Adr
\setbox\Adr\vbox{
\centerline{\sc Christine Bessenrodt}
\vskip18pt
\centerline{Fakult\"at f\"ur Mathematik und Physik,} 
\centerline{Leibniz Universit\"at Hannover,}
\centerline{Welfengarten 1, 30167 Hannover, Germany}
}


\title[Algebra invariants for finite directed graphs   
with relations]{Algebra invariants \\ for finite directed graphs   
with relations}

\author[Christine Bessenrodt]{\box\Adr}

%\address{
%Fakult\"at f\"ur Mathematik und Physik, Leibniz Universit\"at Hannover, 
%Welfengarten 1, 30167 Hannover, Germany}

\begin{abstract}
Finite directed graphs with relations define finite-dim\-en\-sion\-al algebras
given as factor algebras of the corresponding path algebras.
In joint work with T.~Holm, we studied invariants of an associated
generalized Cartan matrix for certain types of algebras where the Cartan
invariants basically count nonzero paths;
the main motivation for this came from the context of
derived module categories and derived invariants.
For the so-called (skewed) gentle algebras we computed explicit
combinatorial formulae for the invariant factors
of the Cartan matrices.
Based on the talk at the FPSAC'05 conference in Taormina, some
of this work is surveyed here, and some further
results on algebras of a different type are presented in the
final section.
\end{abstract}
%\\
%\noindent {\it Mathematics Subject Classification}:
%05E99, 05C38, 05C50, 16G10, 18E30

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\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 54A \rms (2006), Article~B54Ag\hfill}
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\section{Introduction}

Finite directed graphs play an important r\^ole in the representation theory
of finite-dimensional algebras, where they are called {\em quivers}.
For a given quiver~$Q$ and a field $K$ one obtains an algebra $KQ$
by taking all paths in~$Q$, including the trivial paths of length~0
at each vertex, as a $K$-basis, and defines multiplication as induced by
concatenation of paths; this is called the {\em path algebra} to the quiver~$Q$.
The importance of the path algebras
lies in a famous result by Gabriel that any finite-dimensional $K$-algebra
over an algebraically closed field $K$
is Morita equivalent to a factor algebra $KQ/I$, where $I$
is an admissible ideal of $KQ$ (i.e., contained in the  square of
the ideal of $KQ$ generated by the arrows).
Thus such factor algebras $KQ/I$ are central objects of study
in the representation theory of algebras;
this situation is referred to as a {\em quiver with relations}.
There are many interesting representation theoretic properties
of such algebras for which one tries to find ``combinatorial''
methods for computing them.
It is of particular importance to consider
representation theoretic parameters of the algebra which are
invariants for appropriate equivalence classes of algebras.
In recent years a focus in the representation theory of algebras
has been the investigation of derived equivalences of
algebras; this is a homological notion:
two algebras are derived equivalent if their derived module categories
are equivalent.
A lot of progress has recently been made in this very active area.
It is a difficult problem to find invariants of algebras preserved
by derived equivalences;
only a few important representation theoretic
parameters are known to be indeed invariants under derived equivalence,
such as the number of simple modules, the dimension of the
center of the algebra or the dimension of its
Hochschild cohomology groups.
\smallskip

Here, we will discuss invariants of the {\em Cartan matrix} of
a finite-dimensional algebra $A=KQ/I$.
Cartan matrices contain crucial structural information on the algebra,
as their entries are the
multiplicities of simple $A$-modules as composition factors of
projective indecomposable $A$-modules.
It is in general difficult to compute the entries of the Cartan matrix
and some famous conjectures are related to their properties.
The main point to note here  is that the unimodular equivalence
class of the Cartan matrix of a finite dimensional algebra is
invariant under derived equivalence.

From a combinatorial point of view it is important
that for a finite-dimensional algebra $A=KQ/I$ given by a quiver with relations,
the entries of the Cartan matrix can be computed
by considering paths in the quiver $Q$ which are non-zero in the algebra~$A$.
\smallskip

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\markboth{\SMALL CHRISTINE BESSENRODT}{\SMALL ALGEBRA INVARIANTS FOR FINITE DIRECTED GRAPHS WITH RELATIONS}
%
%

The algebras we study here are the (skewed-) gentle algebras which
are defined combinatorially by conditions on the quiver and relations.
Gentle algebras occur naturally in many places in the representation
theory of finite dimensional algebras, in particular in connection
with derived categories. They made their first appearance in 1981~\cite{AH}
when it was shown that the algebras which are derived
equivalent to hereditary algebras of type $\mathbb{A}$ are precisely
the gentle algebras whose underlying undirected graph is a tree.
The algebras which are derived equivalent to hereditary
algebras of type $\tilde{\mathbb{A}}$ are certain gentle
algebras whose underlying graph has exactly one cycle~\cite{AS}.
Only recently it was proved that the class of gentle algebras
has the remarkable property of being closed under derived
equivalence~\cite{SZ}.
For more background on the algebraic context the reader is referred to~\cite{BH}.
\smallskip

In  joint work with T.~Holm~\cite{BH} we have
studied the Cartan matrices of (skewed-)gentle algebras
and some refinements and generalization of this notion;
the starting point of our investigation
was a recent result by
T.~Holm~\cite{TH} giving an explicit combinatorial
formula for the Cartan determinants of gentle algebras.
In~\cite{BH}, these results are improved to a determination of the invariant factors
of the Cartan matrix $C_A$ of a gentle algebra $A=KQ/I$,
and  the formulae are also extended to skewed-gentle algebras.
Sections~2 and~3 survey some of these results,
and (as in the talk at FPSAC'05) we take the opportunity
to include an alternative proof for the fact that the number of
oriented cycles with full zero relations in such a quiver is bounded
by the number of vertices.
The key to the refinement of the combinatorial analysis of the quiver
is to put a weight on the paths according to their lengths instead
of just counting them; in our context, this makes good sense as
the relations on the quiver are homogeneous (in fact, they are even
more special).
Taking an  indeterminate $q$
corresponding to the weight of an arrow,
this gives us a $q$-Cartan  matrix $C_A(q)$
for the algebra; this may also be considered
as a so-called filtered Cartan matrix, counting the multiplicities of the simple
modules in the radical layers of the algebra.
Setting $q=1$ gives the ordinary Cartan matrix~$C_A$.
We have already pointed out that the unimodular equivalence class of the Cartan matrix
of a finite-dimensional algebra is
invariant under derived equivalence.
But unfortunately, not even the determinant
of the $q$-Cartan matrix is in general
an invariant under derived equivalence.

\smallskip

There are further finite-dimensional algebras given by quivers and relations for which
it is possible to determine the $q$-Cartan matrices and obtain nice formulae for
their invariants or at least their determinant.
A further family of algebras associated with cyclic quivers
is considered in the final section~4; here, the $q$-Cartan matrices are special
circulant matrices and we compute their determinants and invariants explicitly.
\smallskip

Finally, in more recent joint work with T.~Holm~\cite{BH-weighted} we have also dealt with
a class of infinite-dimensional algebras which we termed {\em locally gentle algebras};
this class is again defined combinatorially by relaxing the conditions on the
gentle quiver. Also, in this context it is then well motivated to study
more general weights on the quiver.
The combinatorial point of view adds further interest to these quivers, as
for example secant configurations in regular $2n$-polygons may be viewed
as examples of locally gentle quivers.
For details the reader is referred to~\cite{BH-weighted}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Gentle algebras}\label{sec:Gentles}

In this section, we want to describe an extension and refinement of the
result on the determinant of the
Cartan matrix of a gentle algebra from~\cite{TH} obtained in~\cite{BH}
(see~\cite{BH} for details).
\smallskip

First we have to give the definition of gentle algebras.
They form an important subclass of the
class of special biserial algebras which we now define.
\\[1ex]
Let $K$ be an algebraically closed field.
Let~$Q$ be a quiver, i.e., a finite directed graph,
with set of vertices~$Q_0$.
Let $I$ be an admissible
ideal of the path algebra $KQ$, i.e., $I\subseteq J^2$, where
$J$ is  the ideal of $KQ$ generated by the arrows. \\
For a path $p$ in~$Q$ we denote by $s(p)$ its start vertex and $t(p)$ its end vertex.\\
The pair $(Q,I)$ is called {\em special biserial} if the following holds: \\
(i) For any vertex $v\in Q_0$ the set of lengths of the paths starting in $v$
and not being in $I$ is finite.\\
(ii) Each vertex $v\in Q$ is the end point of at most two arrows and the starting point
of at most two arrows.\\
(iii) For every arrow $\alpha$ there is at most one arrow $\beta$ with $t(\alpha)=s(\be)$
and $\alpha\beta \not\in I$, and there is at most one arrow $\gamma$  with
$t(\ga)=s(\alpha)$  and $\gamma\alpha\not\in I$.
\smallskip

A special biserial pair $(Q,I)$ is called {\em gentle},
if furthermore:
\\
(iv) There is a generating set of $I$ (as ideal) consisting of paths of lengths~2.\\
(v) For any arrow $\alpha$ there is at most one arrow $\beta$ with $t(\alpha)=s(\be)$
and $\alpha\beta \in I$, and there is at most one arrow $\gamma$ with
$t(\ga)=s(\alpha)$  and $\gamma\alpha \in I$.
\smallskip

A $K$-algebra $A$ is called {\em gentle} (resp.\ {\em special biserial})
if it is Morita equivalent to an algebra $KQ/I$,
for $(Q,I)$ gentle (resp.\ special biserial).\\
As we assume that the set of vertices~$Q_0$ of~$Q$ is finite,
condition (i) implies that the algebra $KQ/I$ is finite-dimensional.
\\
As mentioned in the introduction, in~\cite{BH-weighted} the notion
of gentle quivers is generalized to locally gentle quivers;
for these quivers, condition (i) need not be satisfied, and thus
the corresponding algebras can be infinite-dimensional.
\smallskip

By condition (iv), the relations for a gentle quiver are homogeneous,
so the length of a non-zero path in $KQ/I$ is well-defined.
Then we define the $q$-Cartan matrix of $A=KQ/I$
as $C_A(q)=(c_{ij}(q))_{i,j\in Q_0}$, where for $i,j\in Q_0$
$$c_{ij}(q)= \sum_{n\geq 0} a_n(i,j) q^{n}$$
with $a_n(i,j)$ the number of paths from $i$ to $j$ which are
non-zero in~$A$ (and different in $A$).
\medskip

{\bf Example.}
In the following picture,
the dotted lines (or arcs) between two arrows or in a loop
correspond to the generating  relations for the quiver, i.e.,
they indicate that composing the corresponding arrows
is zero in the algebra $A=KQ/I$.
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\includegraphics[scale=0.4]{gentle5aex-1.eps}
\end{center}
Here is the $q$-Cartan matrix for this gentle quiver with relations:
$$
C_A(q)=
\left(
\begin{array}{ccccc}
1+q^5&q+q^2+q^7&q+q^4+q^6+q^9&q^3+q^4+q^8&q^2+q^3\\
q^4&1+q^6&q^2+q^5+q^8&q+q^3+q^7&q+q^2\\
0&q&1+q^3&q^2&0\\
q&q^3&q+q^2+q^5&1+q^4&0\\
q^2+q^3&q^4+q^5&q^3+q^4+q^6+q^7&q+q^2+q^5+q^6&1+q
\end{array}
\right)
$$
%%%%%%%%%%%%%%%%%%%%%%

\medskip

The following property does not only hold for gentle algebras but for those
where we have dropped the final condition (v) in the definition of
gentle pairs; in \cite{BH}, it is a useful reduction tool in the proof of the main
result.

\begin{lemma}\label{lem:arrow-removal}
Let $A=KQ/I$ be a  special biserial algebra,
where $I$ is generated by paths of length~2.
Let $\alpha$ be an arrow in~$Q$, not a loop, such that there is no arrow
$\be$ with $s(\al)=t(\be)$ and $\be\alpha\in I$,
or there is no arrow
$\ga$ with $t(\al)=s(\ga)$ and $\al\ga\in I$.
Let~$Q'$ be the quiver obtained from~$Q$ by removing
the arrow $\alpha$, $I'$ the corresponding relation ideal
and $A'=KQ'/I'$.
Then the $q$-Cartan matrices $C_A(q)$ and $C_{A'}(q)$ are
unimodularly equivalent
(over $\Z[q]$).
\end{lemma}


For the combinatorially defined gentle algebras we
can provide an explicit combinatorial description
for a very nice normal form of its $q$-Cartan matrix;
below, we say that a matrix $C$
is unimodularly equivalent (over $\Z[q]$) to
a matrix $D$, if there are matrices $S, T$ with entries in $\Z[q]$
and determinant~1 such that $SCT=D$.

\begin{theorem}\cite{BH}\label{Thm:gentle}
Let $A=KQ/I$ be a  gentle algebra, defined by a gentle pair $(Q,I)$.
Denote by $c_k$ the number of oriented $k$-cycles in~$Q$
with full zero relations.\\
Then the $q$-Cartan matrix $C_A(q)$
is unimodularly equivalent (over $\Z[q]$) to a diagonal matrix with entries
$(1-(-q)^k)$, with multiplicity ${c_k}$, $k\geq 1$,
and all further diagonal entries being~$1$.
\end{theorem}

This result has some immediate nice consequences.
\begin{cor}
Let $A=KQ/I$ be a  gentle algebra,
and denote by $c_k$ the number of oriented $k$-cycles in~$Q$ with full zero relations.
Then the $q$-Cartan matrix $C_A(q)$
has determinant $$\det C_A(q)= \prod_{k \geq 1} (1-(-q)^k)^{c_k} \:.$$
\end{cor}
\medskip

{\bf Example.}
In the example given before, the $q$-Cartan matrix has determinant
$$\det C_A(q) = 1+q+q^3-q^5-q^7-q^8 = (1+q)(1+q^3)(1-q^4)\:.$$
Indeed, the quiver has a loop at vertex~5 (which is a 1-cycle with a zero relation),
a 3-cycle with full zero relations from vertex~1 to 2 to~4 and back to~1, and a 4-cycle
with full zero relations running over 2,5,4,3 and back to~2.
\medskip

For an algebra $A=KQ/I$ as above, let $ec(A)$ and $oc(A)$ be the number
of oriented cycles in~$Q$ with full zero relations of even and odd length, respectively.
Setting $q=1$,  the corollary above immediately implies the main result
from~\cite{TH} which was in fact the starting point of our investigations in~\cite{BH}.
\begin{cor}
Let $A=KQ/I$ be a gentle algebra.
Then for the determinant of its ordinary Cartan
matrix $C_A$ the following holds.
$$\det C_A = \left\{
\begin{array}{ll} 0 & \textrm{if }\: ec(A)>0 \\
                  2^{oc(A)} & \textrm{else}
\end{array}
\right.
$$
\end{cor}

The Theorem also gives the following corollary.
\begin{cor}
Let $A=KQ/I$ be a gentle algebra.
Then there are at most $|Q_0|$ oriented cycles with full zero relations
in the quiver~$Q$.
\end{cor}


Instead of deriving this from the main result (as done in \cite{BH}),
this may be proved directly.
As an illustration, we give this alternative proof here. \\
Clearly the result holds when $|Q_0|=1$. So we assume now that $|Q_0|>1$, and we
prove the result by induction.\\
We may remove all arrows going in or out of a vertex and not having a zero relation at this
vertex, without losing
the property of the quiver with relations being gentle
and without changing the number of vertices and the number of oriented cycles
with full relations.
After this removal, all vertices are of degree~0, 2 or~4, and there is a zero relation
at all vertices of degree~2.
By induction, we may assume that there is no vertex of degree~0 and that the quiver is
connected. If all vertices are of degree~4, then there are paths of arbitrary lengths,
contradicting the property of being gentle.
Hence there is a vertex of degree~2, which then belongs
to a unique oriented cycle with full zero relations.
Removing this vertex and the arrows incident to it reduces the number of vertices as well
as the number of oriented cycles with full zero relations by~1 (note that any arrow
in a gentle quiver belongs to at most one oriented cycle with full zero relations).
Hence the result follows by induction.


\begin{remark}{\rm
Note that in our context  $|Q_0|$ is the number $l(A)$ of simple $A$-modules, which is also
invariant under derived equivalence.
Hence this implies that the Cartan determinant of a gentle algebra $A$ is at most
$2^{l(A)}$.
}\end{remark}

Recall that the property of an algebra  being gentle is invariant under derived equivalence~\cite{SZ}.
Also, we have pointed out earlier that the invariant factors of the ordinary
Cartan matrix $C_A=C_A(1)$ are invariants of the derived equivalence class of the algebra $A=KQ/I$.
Thus we now have some easily computable invariants for gentle algebras
to distinguish the derived equivalence classes.

\begin{cor}
Let $A=KQ/I$ and $A'=KQ'/I'$ be  gentle algebras as above.
If $A$ is derived equivalent to  $A'$, then
$ec(A)=ec(A')$ and $oc(A)=oc(A')$.
\end{cor}


Our new invariants are a quite powerful tool for distinguishing gentle algebras
up to derived equivalence which cannot be separated by the more classical invariants.
An illustration on how to use the invariants to tell non-equivalent gentle algebras apart
is given in~\cite{BH}, where the 9 gentle algebras with two simples and
the 18 gentle algebras with three simples and vanishing Cartan determinant
are discussed in detail.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Skewed-gentle algebras}


Also skewed-gentle algebras are defined combinatorially.
They were introduced in~\cite{GdlP};
for the notation and definition we follow here mostly~\cite{BMM},
but we try to explain how the construction works rather than
stating the technical definitions.
\medskip

We start with a gentle pair $(Q,I)$.
A set $Sp$ of vertices of the quiver~$Q$ is an admissible set of {\em special} vertices
if the quiver with relations obtained from~$Q$ by adding loops with square zero at these
vertices is again gentle; we denote this gentle pair by $(Q^{\text{sp}},I^{\text{sp}})$.
The triple $(Q,Sp,I)$ is then called {\em skewed-gentle}.
\smallskip


We want to point out that the admissibility of the set $Sp$ of special vertices
is both a local as well as a global condition.
Let $v$ be a vertex in the gentle quiver $(Q,I)$;
then we can only add a loop at $v$ if $v$ is of degree 1 or 0
or if it is of degree~2 with a non-loop zero relation.
Hence only vertices of this type are potential special vertices.
But for the choice of an admissible set of special vertices
we also have to take care of the global condition
that after adding all loops, the
pair $(Q^{\text{sp}},I^{\text{sp}})$
still does not have paths
of arbitrary lengths.
\smallskip

Given a skewed-gentle triple $(Q,Sp,I)$, we now construct a new quiver with
relations $(\hat{Q}, \hat{I})$ by doubling the special vertices,
introducing arrows to and from these vertices corresponding to the previous
such arrows and replacing a previous zero relation at the vertices by a
mesh relation.\\
More precisely, we proceed as follows.
The non-special  vertices in~$Q$ are also vertices in the new quiver;
any arrow between non-special vertices as well as corresponding relations
are also kept.
Any special vertex $v\in Sp$ is replaced by two vertices $v^+$ and $v^-$ in the new quiver.
An arrow $a$ in~$Q$ from a non-special vertex $w$ to $v$ (or from $v$ to $w$) will be doubled
to arrows $a^{\pm}:w \to v^{\pm}$ (or $a^{\pm}: v^{\pm} \to w$, resp.) in the new quiver;
an arrow between two special vertices $v,w$ will correspondingly give four arrows
between the pairs $v^{\pm}$ and $w^{\pm}$. We say that these new arrows lie over the
arrow~$a$.
Any relation $ab=0$ where $t(a)=s(b)$ is non-special
gives a corresponding zero relation for paths of length~2 with the same
start and end points lying over~$ab$.
If $v$ is a special vertex of degree~2 in~$Q$, then the corresponding zero relation at~$v$,
say $ab=0$ with $t(a)=v=s(b)$,
is replaced by mesh commutation relations saying that any two paths of length~2
lying over $ab$, having the same start and end points but running over $v^+$ and $v^-$, respectively,
coincide in the factor algebra to the new quiver with relations $(\hat{Q}, \hat{I})$.
\\
We will speak of $(\hat{Q}, \hat{I})$ as a skewed-gentle quiver {\em covering} the
gentle pair $(Q,I)$.
Note that also here the generating relations are homogeneous.
\medskip

A $K$-algebra is then called {\em skewed-gentle}
if it is Morita equivalent to a factor algebra
$K\hat{Q}/\hat{I}$, where $(\hat{Q},\hat{I})$ comes from a skewed-gentle triple $(Q,Sp,I)$
as described above.
\medskip


{\bf Examples.}
(1) We take the gentle quiver~$Q$ as shown below,
with relation ideal $I$ generated by $ab$ and $ba$.
Then we can take $Sp=\{2\}$, i.e., only the vertex~2 is specified as a special vertex.
This gives the skewed-gentle quiver $\hat Q$ shown below, with relation ideal $\hat I$ generated
by $a^+b^+-a^-b^-$, $b^{\pm} a^{\pm}$, $b^{\pm} a^{\mp}$.
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\includegraphics[scale=0.5]{skew-ex1.eps}
\end{center}
\medskip

(2) We take the gentle quiver~$Q$ as shown below, with relation ideal generated
by $ab$. This time we take $Sp=Q_0$, i.e., all vertices are chosen to be special.
This gives the skewed-gentle quiver $\hat Q$ shown below, where for simplicity
all arrows lying over $a$ or $b$, respectively, are also marked $a$ or $b$, respectively,
but for writing down the relations generating
the relation ideal $\hat I$ we will put signs on, so that e.g.\ $a_+^-$ denotes the arrow
going from $1^+$ to $2^-$.
In this notation the generating relations are given by
$a_+^+b_+^+ - a_+^-b_-^+$, $a_+^+b_+^- - a_+^-b_-^-$, $a_-^+b_+^+ - a_-^-b_-^+$,
$a_-^+b_+^- - a_-^-b_-^-$.
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\includegraphics[scale=0.5]{skew-ex2.eps}
\end{center}

\bigskip


Our result on gentle algebras generalizes nicely to skewed-gentle algebras
(see \cite{BH} for details) as follows.


\begin{theorem}\cite{BH}\label{Thm:skew-gentle}
Let $(Q,I)$ be a gentle quiver, $(\hat{Q},\hat{I})$ a covering
skewed-gentle quiver.
Let $\hat A=K\hat Q/\hat I$ be the corresponding skewed-gentle algebra.
Denote by $c_k$ the number of oriented $k$-cycles in $(Q,I)$ with full
zero relations.\\
Then the $q$-Cartan matrix $C_{\hat A}(q)$ is unimodularly equivalent (over $\Z[q]$)
to a diagonal matrix with entries
$1-(-q)^k$, with multiplicity ${c_k}$, $k\geq 1$,
and all further diagonal entries being~$1$.
\end{theorem}



\begin{remark}{\rm
Thus,  the $q$-Cartan matrix $C_A(q)$ for the gentle algebra $A$ to~$(Q,I)$,
and the $q$-Cartan matrix $C_{\hat A}(q)$ for a skewed-gentle cover $\hat A$
are unimodularly equivalent to diagonal matrices which only differ by adding
as many further~$1$'s on the diagonal as there are special vertices chosen in~$Q$;
in particular, with notation as above,
$$\det C_{\hat A}(q) = \det C_A (q)=
\prod_{k \geq 1} (1-(-q)^k)^{c_k} $$
and thus also for the ordinary Cartan matrices
$$\det C_A = \det C_{\hat A}\:.$$
}
\end{remark}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cycles  and circulants}

There are some more types of algebras for which one can determine
the unimodular equivalence class of their $q$-Cartan matrices.
As an example, we consider $q$-Cartan matrices to cyclic
quivers with relations which occur in other interesting contexts.
\\
In this situation, circulants make an appearance, and we first define the
relevant notation.
\smallskip

For
 $x=(x_1,\ldots, x_n)$
the circulant matrix to $x$ is
$$C=
\left(
\begin{array}{ccccc}
x_1 & x_2 & \ldots & x_{n-1} & x_n\\
x_n & x_1 & \ldots & x_{n-2} & x_{n-1}\\
\vdots \\
x_2 & x_3 & \ldots & x_{n} & x_1
\end{array}
\right)\:.
$$


Let $\omega$ be a primitive $n$-th root of unity in $\C$.
Then $C$ has the eigenvalues
$$\sum_{k=1}^n x_k (\omega^j)^{k-1} \:, \: j=1, \ldots ,n.$$
\medskip

Special circulants appear as $q$-Cartan matrices for cyclic
quivers with monomial relations and commutation relations
which are homogeneous of degree~2.
\medskip

{\bf Example.} Here is the corresponding cyclic quiver for the case of 6 vertices:
\begin{center}
\includegraphics[scale=0.5]{cycle.eps}
\end{center}
where we take as generating relations:
$a^2$, $b^2$, $ab-ba$ (at each vertex of the quiver).
\medskip

The $q$-Cartan matrix for such a  cyclic quiver
is just the circulant to $v=(1+q^2,q, 0 \ldots, 0, q)$.
We will describe a more precise result on these $q$-Cartan matrices below,
but as it is easy to do we compute here the determinant.
Note that as in the case of gentle quivers we have a contribution $1-(-q)^n$
for each of the two oriented cycles with full zero relations in the quiver.

\begin{theorem}
Let $q$ be an indeterminate, $v=(1+q^2,q, 0 \ldots, 0, q)$ 
{\em(}of length~$n${\em)},
and $C(q)$ the circulant to~$v$.
Then we have
$$\det C(q) = (1-(-q)^n)^2\:.$$
\end{theorem}

\begin{proof}
By the above, we have
$$
\begin{array}{rcl}
\det C(q) & = &
\displaystyle \prod_{j=1}^n (1+q^2+q\omega^j+q(\omega^j)^{n-1})
=
\displaystyle \prod_{j=1}^n (1+q^2+q(\omega^j+\overline{\omega^j}))
\\& = &
\displaystyle \prod_{j=1}^n (q+ \omega^j)(q+\overline{\omega^j})
=
\displaystyle (\prod_{j=1}^n (q+ \omega^j))^2\\
\end{array}
\:.$$
Now $-\omega^j$ is a zero of $q^n-1$, if $n$ is even
and a zero of $q^n+1$, if $n$ is odd, for all $j$, hence we
have the assertion.
\end{proof}


In fact, it is not hard to transform
the circulant $C(q)$ to $v=(1+q^2,q, 0 \ldots, 0, q)$
into a better form:

\begin{theorem}
Assume $n\geq 3$, and let $C(q)$ be as above.\\
{\em(i)}
Over $\Z[q]$, we can transform $C(q)$ unimodularly
into the form
$$
\left(
\begin{array}{ccc}
E_{n-2} & 0 & 0\\
0 & 1-(-q)^n & \big(\displaystyle\sum_{j=1}^{n-1}q^{2j-1}\big) - (-q)^{n-1} \\
0 & 0 & 1-(-q)^n
\end{array}
\right)
$$
where $E_{n-2}$ is the identity matrix of type $n-2$.
\\
{\em(ii)}
Over~$\Q[q]$, we have the following unimodular equivalences:
\\
For $n$ odd, \quad
$\displaystyle C(q)\sim \diag(1^{n-2},\sum_{j=0}^{n-1}(-q)^j, q^{n+1}+q^n+q+1)\:. $
\\[1ex]
For $n$ even, \quad
$\displaystyle C(q)\sim \diag(1^{n-2},\sum_{j=0}^{(n-2)/2}q^{2j}, q^{n+2}-q^n-q^2+1)\:. $
\end{theorem}



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\end{document}
