Demonet Quiver in Skew Group Algebras

• Demonet quiver is a combinatorial model that presents the Morita reduced skew group algebra as the path algebra of a modified quiver stemming from a finite group action.
• It defines vertices as pairs (i, U) and constructs arrow spaces via Hom-spaces, linking orbifold choices and local module structure in a clear representation-theoretic framework.
• Intertwiner formulas and nondegenerate pairings yield a unique expansion for algebra elements as linear combinations of paths, offering explicit computational methods.

Searching arXiv for the cited paper and closely related records. arXiv search query: (Meur, 2018) 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 $G$0 of a new quiver $G$1 (Meur, 2018). In this form, the Demonet quiver gives a concrete combinatorial model for the Morita reduced algebra, while explicit formulas identify arbitrary elements of $G$2 as $G$3-linear combinations of paths in $G$4.

1. Algebraic setting and idempotent reduction

Let $G$5 be a finite quiver, let $G$6 be the vertex-idempotent algebra, and let $G$7 be the $G$8-bimodule spanned by the arrows of $G$9, so that $kQ$0. The finite group $kQ$1 acts by automorphisms on $kQ$2, and the resulting skew group algebra is

$kQ$3

(Meur, 2018).

For each vertex $kQ$4, the stabiliser is

$kQ$5

and $kQ$6 denotes the corresponding primitive idempotent. One fixes a complete set of irreducible $kQ$7-modules $kQ$8, and for each $kQ$9 chooses a primitive idempotent $Q$0 such that

$Q$1

With $Q$2 a choice of representatives of the $Q$3-orbits of vertices, the idempotent used in the reduction is

$Q$4

Reiten–Riedtmann and Demonet show that $Q$5 and $Q$6 are Morita equivalent, and that under the stated assumption on the chosen simples $Q$7, there is a canonical algebra isomorphism

$Q$8

(Meur, 2018).

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$9 is a pair $k$0 where $k$1 and $k$2 (Meur, 2018). Thus each orbit representative $k$3 is refined by the choice of an irreducible representation of its stabiliser.

For two such vertices $k$4 and $k$5, one first forms the $k$6-module

$k$7

where $k$8 denotes the conjugate $k$9-module $\operatorname{char}(k)\nmid |G|$0. The vector space of arrows

$\operatorname{char}(k)\nmid |G|$1

has a natural $\operatorname{char}(k)\nmid |G|$2-basis given by

$\operatorname{char}(k)\nmid |G|$3

Equivalently, the number of arrows is

$\operatorname{char}(k)\nmid |G|$4

Concretely, one chooses a $\operatorname{char}(k)\nmid |G|$5-basis of $\operatorname{char}(k)\nmid |G|$6 and declares each basis element to be an arrow of $\operatorname{char}(k)\nmid |G|$7 (Meur, 2018).

The construction can be summarized as follows.

Component	Definition
Vertices	Pairs $\operatorname{char}(k)\nmid |G|$8 with $\operatorname{char}(k)\nmid |G|$9 and $R=(kQ)*G.$0
Arrow source/target	From $R=(kQ)*G.$1 to $R=(kQ)*G.$2
Arrow space	$R=(kQ)*G.$3

This description makes the representation theory of the stabilisers intrinsic to the quiver itself. A plausible implication is that $R=(kQ)*G.$4 encodes, at the level of vertices and arrow multiplicities, both the orbit structure of $R=(kQ)*G.$5 on $R=(kQ)*G.$6 and the local module-theoretic data of the stabiliser subgroups.

3. Canonical identification of $R=(kQ)*G.$7 with the basic algebra

The isomorphism

$R=(kQ)*G.$8

is canonical in the sense described in the source: each arrow

$R=(kQ)*G.$9

of $\tilde e\in (kQ)*G$0 is sent to

$\tilde e\in (kQ)*G$1

(Meur, 2018).

In this form, the basic algebra $\tilde e\in (kQ)*G$2 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 $\tilde e\in (kQ)*G$3, the stabilisers $\tilde e\in (kQ)*G$4, the sets $\tilde e\in (kQ)*G$5, and the idempotents $\tilde e\in (kQ)*G$6 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\in (kQ)*G$7 as linear combinations of paths in $\tilde e\in (kQ)*G$8 (Meur, 2018). This is done via the algebra of intertwiners and a monoidal-category formalism.

For each path $\tilde e\in (kQ)*G$9 in $eRe=\tilde e\,(kQ*G)\,\tilde e$0 of length $eRe=\tilde e\,(kQ*G)\,\tilde e$1, one associates an intertwiner

$eRe=\tilde e\,(kQ*G)\,\tilde e$2

obtained by iterated $eRe=\tilde e\,(kQ*G)\,\tilde e$3-product of the individual-arrow maps $eRe=\tilde e\,(kQ*G)\,\tilde e$4. Dual to $eRe=\tilde e\,(kQ*G)\,\tilde e$5 is a second intertwiner $eRe=\tilde e\,(kQ*G)\,\tilde e$6 relative to the dual bimodule $eRe=\tilde e\,(kQ*G)\,\tilde e$7, chosen so that

$eRe=\tilde e\,(kQ*G)\,\tilde e$8

Theorem 2.12 asserts that the map

$eRe=\tilde e\,(kQ*G)\,\tilde e$9

is an algebra isomorphism, where $R$0 is the algebra of intertwiners. Moreover, every intertwiner $R$1 of degree $R$2 has the unique expansion

$R$3

and the coefficients are computed by nondegenerate pairings in the monoidal category of $R$4-bimodules.

Composing with the inverse of Demonet’s identification yields an algebra isomorphism

$R$5

and therefore any element $R$6 admits the explicit path expansion

$R$7

Hence the coefficient of a basis path $R$8 is exactly the scalar

$R$9

This gives a computational answer to the Morita reduction problem: not only is $G$00 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 $G$01 has no further relations beyond those of a free path algebra, all paths in $G$02 are linearly independent (Meur, 2018). There is no mesh relation or zero relation at the level of $G$03 itself.

The source emphasizes that the only relations arise indirectly from the $G$04-action in $G$05. These appear through coefficients $G$06 describing how $G$07 permutes paths of $G$08, and these coefficients influence the decomposition of a $G$09-skewed cycle in the $G$10 basis. Thus the combinatorial freedom of the path algebra is preserved in $G$11, while the group action re-enters through the expansion coefficients of elements coming from $G$12.

Two examples in the source illustrate the construction. In one example, $G$13 and $G$14 swaps $G$15 while fixing a new vertex $G$16. The resulting quiver has decorated vertices such as $G$17, $G$18, and $G$19, and the arrow multiplicities are determined by the four $G$20-modules $G$21 computed there (Meur, 2018).

In another example, $G$22 is the cyclic quiver on $G$23 vertices, $G$24 is the dihedral group, and one considers the $G$25-cycle

$G$26

The formulas yield

$G$27

summed over all length-$G$28 oriented cycles $G$29 in $G$30. No further local relation is imposed (Meur, 2018).

6. Morita equivalence, hereditary structure, and graded form

The Morita reduction proceeds in two steps. First, the skew group algebra $G$31 is hereditary and Morita equivalent to its basic algebra

$G$32

Second, one reduces further by the idempotent $G$33, so that

$G$34

is again Morita equivalent to $G$35 (Meur, 2018). Demonet’s theorem then identifies this final basic algebra with the path algebra of $G$36.

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 $G$37 agrees with the grading by tensor length in $G$38 (Meur, 2018). This compatibility is structurally important because it preserves the tensor-length filtration inherited from the presentation $G$39.

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 $G$40 and the isomorphism

$G$41

Le Meur develops the explicit formulas, through intertwiners, monoidal-category operations, and nondegenerate pairings, that decompose arbitrary elements of $G$42 in the path basis of $G$43 (Meur, 2018).

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.