---
title: Demonet Quiver in Skew Group Algebras
url: https://www.emergentmind.com/topics/demonet-quiver
type: topic
---

# Demonet Quiver in Skew Group Algebras

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arXiv search query: 1810.12612
The **Demonet quiver** $Q_G$ is the quiver that appears in the Morita reduction of a skew group algebra of a path algebra. For a finite group $G$ acting by automorphisms on the path algebra $kQ$ of a finite quiver $Q$, with $k$ algebraically closed and $\operatorname{char}(k)\nmid |G|$, one considers the skew group algebra
$$
R=(kQ)*G.
$$
Reiten–Riedtmann and Demonet show that there exists an idempotent $\tilde e\in (kQ)*G$ such that the basic algebra
$$
eRe=\tilde e\,(kQ*G)\,\tilde e
$$
is Morita equivalent to $R$ and canonically isomorphic to the path algebra $kQ_G$ of a new quiver $Q_G$ [1810.12612]. In this form, the Demonet quiver gives a concrete combinatorial model for the Morita reduced algebra, while explicit formulas identify arbitrary elements of $eRe$ as $k$-linear combinations of paths in $Q_G$.

## 1. Algebraic setting and idempotent reduction

Let $Q$ be a finite quiver, let $S=k^{Q_0}$ be the vertex-idempotent algebra, and let $M$ be the $S$-bimodule spanned by the arrows of $Q$, so that $kQ\cong T_S(M)$. The finite group $G$ acts by automorphisms on $kQ$, and the resulting skew group algebra is
$$
R=(kQ)*G
$$
[1810.12612].

For each vertex $i\in Q_0$, the stabiliser is
$$
G_i=\{\,g\in G\mid g\cdot i=i\},
$$
and $e_i\in S$ denotes the corresponding primitive idempotent. One fixes a complete set of irreducible $kG_i$-modules $\mathrm{irr}(G_i)$, and for each $U\in \mathrm{irr}(G_i)$ chooses a primitive idempotent $\varepsilon_U\in kG_i$ such that
$$
U\cong kG_i\,\varepsilon_U.
$$

With $[G\backslash Q_0]$ a choice of representatives of the $G$-orbits of vertices, the idempotent used in the reduction is
$$
\tilde e=\sum_{\,i\in[G\backslash Q_0]\,,\;U\in\mathrm{irr}(G_i)}
\bigl(e_i*\varepsilon_U\bigr)\in (kQ)*G.
$$
Reiten–Riedtmann and Demonet show that $(kQ)*G$ and $\tilde e\,(kQ*G)\,\tilde e$ are Morita equivalent, and that under the stated assumption on the chosen simples $U$, there is a canonical algebra isomorphism
$$
kQ_G\xrightarrow{\sim}\tilde e\,(kQ*G)\,\tilde e
$$
[1810.12612].

This establishes the Demonet quiver not merely as an auxiliary combinatorial gadget, but as the literal path-algebra presentation of the Morita reduced algebra.

## 2. Vertex set and arrow spaces

A vertex of $Q_G$ is a pair $(i,U)$ where $i\in [G\backslash Q_0]$ and $U\in \mathrm{irr}(G_i)$ [1810.12612]. Thus each orbit representative $i$ is refined by the choice of an irreducible representation of its stabiliser.

For two such vertices $(i,U)$ and $(j,V)$, one first forms the $G_i$-module
$$
M(i,j;V)
=
\bigoplus_{y\,\in\,[G/G_j]}\;
{}_iM_{y\!j}\,\otimes_{k}\,(yV),
$$
where $yV$ denotes the conjugate $kG_{y\!j}$-module $y\otimes_{kG_j}V$. The vector space of arrows
$$
(i,U)\longrightarrow (j,V)
$$
has a natural $k$-basis given by
$$
\Hom_{kG_i}\bigl(U,M(i,j;V)\bigr).
$$
Equivalently, the number of arrows is
$$
\dim_k\Hom_{G_i}\bigl(U,M(i,j;V)\bigr).
$$
Concretely, one chooses a $k$-basis of $\Hom_{G_i}(U,M(i,j;V))$ and declares each basis element to be an arrow of $Q_G$ [1810.12612].

The construction can be summarized as follows.

| Component | Definition |
|---|---|
| Vertices | Pairs $(i,U)$ with $i\in [G\backslash Q_0]$ and $U\in \mathrm{irr}(G_i)$ |
| Arrow source/target | From $(i,U)$ to $(j,V)$ |
| Arrow space | $\Hom_{kG_i}\bigl(U,M(i,j;V)\bigr)$ |

This description makes the representation theory of the stabilisers intrinsic to the quiver itself. A plausible implication is that $Q_G$ encodes, at the level of vertices and arrow multiplicities, both the orbit structure of $G$ on $Q_0$ and the local module-theoretic data of the stabiliser subgroups.

## 3. Canonical identification of $kQ_G$ with the basic algebra

The isomorphism
$$
kQ_G\;\xrightarrow{\sim}\;\tilde e\,(kQ*G)\,\tilde e
$$
is canonical in the sense described in the source: each arrow
$$
\alpha\colon (i,U)\to (j,V)
$$
of $Q_G$ is sent to
$$
\alpha(\varepsilon_U)\in \tilde e\,(kQ*G)\,\tilde e
$$
[1810.12612].

In this form, the basic algebra $eRe=\tilde eRe$ is literally the path algebra of the Demonet quiver. The statement is stronger than a mere existence theorem for a Morita equivalent quiver algebra: it identifies the reduced algebra explicitly and functorially through the chosen idempotent and the decorated vertex set.

The data entering the quiver are not arbitrary. The choice of orbit representatives $[G\backslash Q_0]$, the stabilisers $G_i$, the sets $\mathrm{irr}(G_i)$, and the idempotents $\varepsilon_U$ together determine the vertex set and the canonical embedding of arrows into the reduced skew group algebra. This suggests that the Demonet quiver is best understood as the representation-theoretic refinement of the orbit quiver obtained from the group action.

## 4. Intertwiners and explicit path expansions

A central contribution of the cited work is an explicit procedure for decomposing arbitrary elements of $\tilde e\,(kQ*G)\,\tilde e$ as linear combinations of paths in $Q_G$ [1810.12612]. This is done via the algebra of intertwiners and a monoidal-category formalism.

For each path $\gamma$ in $Q_G$ of length $n$, one associates an intertwiner
$$
f_\gamma\in
\Hom_{kG_{\,i_0}}\!\bigl(U_0,\;M(i_0,i_1,\dots,i_n;\,U_n)\bigr)
\subset \tilde e\,(kQ*G)\,\tilde e,
$$
obtained by iterated $\circledast$-product of the individual-arrow maps $f_t$. Dual to $f_\gamma$ is a second intertwiner $\varphi_\gamma$ relative to the dual bimodule $M^*$, chosen so that
$$
\bigl(f_{\gamma'}\mid\varphi_\gamma\bigr)=\delta_{\gamma',\gamma}.
$$

Theorem 2.12 asserts that the map
$$
kQ_G\longrightarrow \intw^{\mathrm{op}},
\qquad
\gamma\mapsto f_\gamma
$$
is an algebra isomorphism, where $\intw$ is the algebra of intertwiners. Moreover, every intertwiner $f\in \intw$ of degree $n$ has the unique expansion
$$
f
=
\sum_{\gamma\,:\,\ell(\gamma)=n}
\bigl(f\mid\varphi_\gamma\bigr)\,
f_\gamma,
$$
and the coefficients are computed by nondegenerate pairings in the monoidal category of $A$-bimodules.

Composing with the inverse of Demonet’s identification yields an algebra isomorphism
$$
\Xi\colon \tilde e\,(kQ*G)\,\tilde e\to \intw^{\mathrm{op}},
$$
and therefore any element $\theta\in \tilde e\,(kQ*G)\,\tilde e$ admits the explicit path expansion
$$
\theta
=
\sum_{\gamma}
\bigl(\Xi(\theta)\,\bigm|\;\varphi_\gamma\bigr)\;\gamma
\quad\in\; kQ_G.
$$
Hence the coefficient of a basis path $\gamma$ is exactly the scalar
$$
\bigl(\Xi(\theta)\mid \varphi_\gamma\bigr).
$$

This gives a computational answer to the Morita reduction problem: not only is $eRe$ identified with a path algebra, but arbitrary elements of the reduced algebra can be written explicitly in the path basis by a pairing formula.

## 5. Path structure, linear independence, and examples

Because $eRe\cong kQ_G$ has no further relations beyond those of a free path algebra, all paths in $Q_G$ are linearly independent [1810.12612]. There is no mesh relation or zero relation at the level of $kQ_G$ itself.

The source emphasizes that the only relations arise indirectly from the $G$-action in $kQ*G$. These appear through coefficients $\chi_{g,\gamma}$ describing how $g\in G$ permutes paths of $Q$, and these coefficients influence the decomposition of a $G$-skewed cycle in the $Q_G$ basis. Thus the combinatorial freedom of the path algebra is preserved in $kQ_G$, while the group action re-enters through the expansion coefficients of elements coming from $(kQ)*G$.

Two examples in the source illustrate the construction. In one example, $Q_0=\{\bar0,\bar1\}$ and $G\cong \mathbb Z/2$ swaps $\bar0\leftrightarrow \bar1$ while fixing a new vertex $\bullet$. The resulting quiver has decorated vertices such as $(\bullet,\rho_0)$, $(\bullet,\rho_1)$, and $(\bar0,k_{\mathrm{triv}})$, and the arrow multiplicities are determined by the four $G_i$-modules $M(i,j;V)$ computed there [1810.12612].

In another example, $Q$ is the cyclic quiver on $5$ vertices, $G=\mathrm D_{10}$ is the dihedral group, and one considers the $5$-cycle
$$
w
=
x_{0\!,1}\,x_{1\!,2}\,x_{2\!,3}\,x_{3\!,4}\,x_{4\!,0}
-\bigl(\text{its reverse}\bigr)
\in (kQ)*G.
$$
The formulas yield
$$
\tilde e\,w\,\tilde e
=
-2\sum_{\gamma}\gamma,
$$
summed over all length-$5$ oriented cycles $\gamma$ in $Q_G$. No further local relation is imposed [1810.12612].

## 6. Morita equivalence, hereditary structure, and graded form

The Morita reduction proceeds in two steps. First, the skew group algebra $(kQ)*G$ is hereditary and Morita equivalent to its basic algebra
$$
\hat e\,(kQ*G)\,\hat e,
\qquad
\hat e=\sum_{i\in [G\backslash Q_0]} e_i.
$$
Second, one reduces further by the idempotent $\tilde e\le \hat e$, so that
$$
\tilde e\,(kQ*G)\,\tilde e
$$
is again Morita equivalent to $\hat e\,(kQ*G)\,\hat e$ [1810.12612]. Demonet’s theorem then identifies this final basic algebra with the path algebra of $Q_G$.

The graded setting behaves compatibly with the ungraded one: all of the above remains valid with graded Morita equivalences, and the degree of an arrow in $Q_G$ agrees with the grading by tensor length in $kQ*G$ [1810.12612]. This compatibility is structurally important because it preserves the tensor-length filtration inherited from the presentation $kQ\cong T_S(M)$.

The principal references named in the source delineate the development of the subject. Reiten–Riedtmann provide the hereditary skew-group-algebra framework and the initial Morita reduction. Demonet gives the explicit quiver $Q_G$ and the isomorphism
$$
kQ_G\cong \tilde e\,(kQ*G)\,\tilde e.
$$
Le Meur develops the explicit formulas, through intertwiners, monoidal-category operations, and nondegenerate pairings, that decompose arbitrary elements of $eRe$ in the path basis of $Q_G$ [1810.12612].

Taken together, these results present the Demonet quiver as the exact path-algebra realization of the Morita reduced skew group algebra, with a fully explicit translation between algebra elements and quiver paths.

Source: https://www.emergentmind.com/topics/demonet-quiver