[Paper Review] On McKay Quiver and Covering Spaces

Jin Yun Guo|arXiv (Cornell University)|Feb 9, 2010

Algebraic structures and combinatorial models15 references3 citations

TL;DR

This paper establishes that the McKay quiver of a finite subgroup $G$ of $\mathrm{GL}(m,\mathbb{C})$ can be constructed as a regular covering of the McKay quiver of its $\mathrm{SL}(m,\mathbb{C})$-kernel $N$, under the condition that all irreducible characters of $N$ are extendible to $G$. It further shows that embedding $G$ into $\mathrm{SL}(m+1,\mathbb{C})$ via a natural map adds a loop at each vertex via Nakayama translation, yielding new McKay quivers. These constructions explain known quivers in representation theory and physics, including $\tilde{A}_n$ and D-brane quivers.

ABSTRACT

In this paper, we study the relationship between the McKay quivers of a finite subgroups $G$ of special linear groups general linear groups, via some natural extension and embedding. We show that the McKay quiver of certain extension of a finite subgroup $G$ of $\mathrm{SL}(m,\mathbb C)$ in $\mathrm{GL}(m,\mathbb C)$ is a regular covering of the McKay quiver of $G$, and when embedding $G$ in a canonical way into $\mathrm{GL}(m-1,\mathbb C)$, the new McKay quiver is obtained by adding an arrow from the Nakayama translation of $i$ back to $i$ for each $i$. We also show that certain interesting examples of McKay quivers are obtained in these two ways.

Motivation & Objective

• To clarify the relationship between McKay quivers of finite subgroups of $\mathrm{GL}(m,\mathbb{C})$ and $\mathrm{SL}(m,\mathbb{C})$.
• To show that the McKay quiver of a $\mathrm{GL}(m,\mathbb{C})$ extension of a finite $\mathrm{SL}(m,\mathbb{C})$ subgroup is a regular covering of the original quiver.
• To demonstrate that embedding a $\mathrm{GL}(m,\mathbb{C})$ group into $\mathrm{SL}(m+1,\mathbb{C})$ generates new McKay quivers by adding loops via Nakayama translation.
• To provide a unified construction for known quivers in representation theory and string theory, such as $\tilde{A}_n$ and D-brane quivers.
• To establish that the universal cover of a McKay quiver can be realized through these constructions, linking representation theory and topology.

Proposed method

• Use the determinant homomorphism $\mathrm{det}: G \to \mathbb{C}^*$ to show $G/N$ is cyclic, where $N = G \cap \mathrm{SL}(m,\mathbb{C})$.
• Apply the condition that all irreducible characters of $N$ are extendible to $G$ to construct a regular covering map from the McKay quiver of $G$ to that of $N$ with group $G/N$.
• Define a natural embedding of $G \leq \mathrm{GL}(m,\mathbb{C})$ into $\mathrm{SL}(m+1,\mathbb{C})$ via a map $f$ that shifts the representation space.
• Show that the new McKay quiver is obtained by adding an arrow from the Nakayama translation of each vertex $i$ back to $i$.
• Use the universal covering space construction to relate the representation theory of skew group algebras over exterior algebras.
• Apply these constructions to generate known quivers, including double quivers of $\tilde{A}_n$ and quivers from D-brane physics.

Experimental results

Research questions

• RQ1 Under what conditions is the McKay quiver of a finite subgroup $G \leq \mathrm{GL}(m,\mathbb{C})$ a regular covering of the McKay quiver of its $\mathrm{SL}(m,\mathbb{C})$-kernel $N$?
• RQ2 How does embedding a finite subgroup of $\mathrm{GL}(m,\mathbb{C})$ into $\mathrm{SL}(m+1,\mathbb{C})$ modify its McKay quiver?
• RQ3 What role does Nakayama translation play in constructing McKay quivers for $\mathrm{GL}(m,\mathbb{C})$ subgroups?
• RQ4 Can the double quiver of an affine Dynkin diagram $\tilde{A}_n$ be systematically constructed from McKay quivers of $\mathrm{GL}(1,\mathbb{C})$ subgroups?
• RQ5 How do these constructions explain the McKay quivers arising in D-brane physics, such as those for $\mathbb{P}^{1,1,1,1,2}$?

Key findings

• The McKay quiver of $G \leq \mathrm{GL}(m,\mathbb{C})$ is a regular covering of the McKay quiver of $N = G \cap \mathrm{SL}(m,\mathbb{C})$ when all irreducible characters of $N$ extend to $G$, with covering group $G/N$.
• When $G \leq \mathrm{GL}(m,\mathbb{C})$ is embedded into $\mathrm{SL}(m+1,\mathbb{C})$ via the natural map $f$, the resulting McKay quiver is obtained by adding a loop from each vertex $i$ to its Nakayama translation $\tau(i)$.
• The double quiver of the affine Dynkin diagram $\tilde{A}_{n-1}$ arises as the McKay quiver of a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ constructed by extending a trivial group in $\mathrm{GL}(1,\mathbb{C})$ and embedding into $\mathrm{SL}(2,\mathbb{C})$.
• The McKay quiver for $\mathbb{P}^{1,1,1,1,2}$ in D-brane theory is constructed by extending the trivial group in $\mathrm{SL}(4,\mathbb{C})$ with a cyclic group of order 6 in $\mathrm{GL}(4,\mathbb{C})$, then embedding into $\mathrm{SL}(4,\mathbb{C})$ to add loops via Nakayama translation.
• The only 3-McKay quivers with 4 vertices for finite subgroups of $\mathrm{GL}(m,\mathbb{C})$ are obtained by applying the Nakayama translation construction to the double quiver of $\tilde{A}_1$.
• The universal cover of a McKay quiver can be constructed via these methods, yielding a simply connected quiver that explains earlier results in the $m=2$ case.