$G = \mu_2 = \{1, -1\}$ acting on $\mathbb{A}^2$ by $(x, y) \mapsto (-x, -y)$. The invariant ring is $k[x^2, xy, y^2] \cong k[u, v, w]/(uw - v^2)$. The quotient $\mathbb{A}^2/\mu_2$ is the quadric cone, with a node at the origin.

The minimal resolution is the blowup at the origin: one exceptional $(-2)$-curve $E \cong \mathbb{P}^1$. The dual graph is a single vertex: $A_1$.

The stack $[\mathbb{A}^2/\mu_2]$ is smooth of dimension 2, with a single $\mu_2$-stacky point at the origin.

$G = \mu_3$ acting by $\zeta \cdot (x, y) = (\zeta x, \zeta^{-1} y)$ where $\zeta^3 = 1$. The invariant ring is $k[x^3, xy, y^3] \cong k[u, v, w]/(uw - v^3)$. The minimal resolution has two exceptional $(-2)$-curves $E_1, E_2$ with $E_1 \cdot E_2 = 1$: the $A_2$ Dynkin diagram.

$G = BD_8$ (the quaternion group $Q_8$, which is the binary dihedral group of order 8), acting on $\mathbb{A}^2$. The quotient singularity is $D_4$: $x^2 + y^2 z + z^3 = 0$.

The minimal resolution has four exceptional $(-2)$-curves meeting in the $D_4$ pattern: one central curve meeting three others. The $BD_8$ group has 5 irreducible representations (4 of dimension 1 and 1 of dimension 2), and by the McKay correspondence, these correspond to vertices of the extended $D_4$ diagram (with the trivial representation corresponding to the affine node).

$G = \tilde{A}_5$ (binary icosahedral, order 120), the double cover of the icosahedral group $A_5 \subset \mathrm{SO}(3)$ lifted to $\mathrm{SL}(2)$. The quotient is $x^2 + y^3 + z^5 = 0$ with a minimal resolution having 8 exceptional $(-2)$-curves in the $E_8$ pattern. The group $\tilde{A}_5$ has 9 irreducible representations, matching the 9 nodes of the extended $E_8$ diagram.

This is the most "complex" surface singularity: the resolution has the largest exceptional fiber, and the Milnor number is $\mu = 8$.

$G = \mu_3$ with irreducible representations $\rho_0 = \mathbf{1}$, $\rho_1$, $\rho_2$ (the three cube roots of unity characters). The natural representation is $Q = \rho_1 \oplus \rho_2$. Computing $Q \otimes \rho_i$:

• $Q \otimes \rho_0 = \rho_1 \oplus \rho_2$
• $Q \otimes \rho_1 = \rho_2 \oplus \rho_0$
• $Q \otimes \rho_2 = \rho_0 \oplus \rho_1$

The McKay quiver is: $0 - 1 - 2 - 0$ (a cycle), which is the extended $\tilde{A}_2$ diagram. Removing vertex 0 gives $1 - 2$, the $A_2$ diagram. The two exceptional curves $E_1, E_2$ in the resolution correspond to $\rho_1, \rho_2$.

$G = \mu_n$ with characters $\rho_0, \ldots, \rho_{n-1}$. The McKay quiver is the extended $\tilde{A}_{n-1}$ diagram (a cycle of $n$ vertices). The $n - 1$ nontrivial representations correspond to the $n - 1$ exceptional $(-2)$-curves in the minimal resolution of the $A_{n-1}$ singularity. Each representation has dimension 1, matching the fact that all components of the minimal imaginary root of $\tilde{A}_{n-1}$ are 1.

$G = Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$. The irreducible representations are:

• $\rho_0 = \mathbf{1}$ (trivial), $\dim = 1$
• $\rho_1, \rho_2, \rho_3$ (three non-trivial 1-dimensional), $\dim = 1$ each
• $\rho_4$ (the natural 2-dimensional representation), $\dim = 2$

The McKay graph: $\rho_1, \rho_2, \rho_3$ each connect to $\rho_4$, and $\rho_0$ connects to $\rho_4$. This is exactly the extended $\tilde{D}_4$ diagram. Removing $\rho_0$ gives $D_4$: one central node ($\rho_4$, corresponding to the central exceptional curve of self-intersection $-2$) connected to three outer nodes ($\rho_1, \rho_2, \rho_3$, the three "legs" of $D_4$).

The dimensions $(1, 1, 1, 1, 2)$ are the coefficients of the null vector of $\tilde{D}_4$: $\rho_0 + \rho_1 + \rho_2 + \rho_3 + 2\rho_4 = 0$ in the root lattice.

$G = \mu_2 \subset \mathrm{SL}(2)$. The $G$-Hilbert scheme $Y = \operatorname{Hilb}^{\mu_2}(\mathbb{A}^2)$ is the minimal resolution of $\mathbb{A}^2/\mu_2$ (the blowup at the node). The BKR equivalence gives: $D^b(\operatorname{Coh}^{\mu_2}(\mathbb{A}^2)) \cong D^b(\operatorname{Coh}(Y)).$

Under this equivalence, the non-trivial representation $\rho_1$ of $\mu_2$ corresponds to the line bundle $\mathcal{O}_Y(-E)$ (twisted by the exceptional divisor), and the regular representation corresponds to the structure sheaf $\mathcal{O}_Y$.

For $G = \mu_3 \subset \mathrm{SL}(3)$ acting by $\zeta \cdot (x,y,z) = (\zeta x, \zeta y, \zeta z)$, the quotient $\mathbb{A}^3/\mu_3$ has a terminal singularity. The $G$-Hilbert scheme $Y = \operatorname{Hilb}^{\mu_3}(\mathbb{A}^3)$ is a crepant resolution (it exists because $G$ is abelian), and BKR gives: $D^b(\operatorname{Coh}^{\mu_3}(\mathbb{A}^3)) \cong D^b(\operatorname{Coh}(Y)).$ The resolution $Y$ has two exceptional divisors, corresponding to the two non-trivial representations of $\mu_3$.

The $A_{n-1}$ singularity $\operatorname{Spec} k[u,v,w]/(uv - w^n)$ is resolved by a sequence of blowups, producing $n-1$ exceptional $(-2)$-curves $E_1, \ldots, E_{n-1}$ in a chain: $E_1 - E_2 - \cdots - E_{n-1}$ with intersection matrix $E_i \cdot E_j = -2\delta_{ij} + \delta_{|i-j|,1}$ (the negative of the $A_{n-1}$ Cartan matrix).

Alternatively, the resolution can be described as the toric resolution associated to subdividing the cone $\sigma = \operatorname{Cone}(e_1, e_1 + ne_2)$ by inserting rays $e_1 + ke_2$ for $k = 1, \ldots, n-1$.

The group $G = \mu_2$ acting on $\mathbb{A}^4$ by $(x_1, x_2, x_3, x_4) \mapsto (-x_1, -x_2, -x_3, -x_4)$ gives $G \subset \mathrm{SL}(4)$ (the determinant of $-I_4$ is $(-1)^4 = 1$). The quotient $\mathbb{A}^4/\mu_2$ has a singularity that admits no crepant resolution: any resolution must introduce discrepancy. However, the quotient stack $[\mathbb{A}^4/\mu_2]$ is still smooth.

This motivates the use of stacks as "non-commutative resolutions" or "stacky resolutions" in higher dimensions.

The McKay correspondence has a K-theoretic formulation: there is an isomorphism $K^G_0(\mathbb{A}^n) \cong K_0(Y)$ for a crepant resolution $Y \to \mathbb{A}^n/G$. In the stacky language: $K_0([\mathbb{A}^n/G]) \cong K_0(Y).$

For $G \subset \mathrm{SL}(2)$: $K^G_0(\mathbb{A}^2) \cong K^G_0(\mathrm{pt}) = R(G)$ (the representation ring), and $K_0(Y)$ is free abelian of rank $|G|$ (the number of irreducible representations). The basis of $R(G)$ given by irreducible representations maps to a basis of $K_0(Y)$ given by the exceptional divisors plus the structure sheaf.

The classification of terminal quotient singularities in dimension 3 (by Mori and Morrison--Stevens): these are of the form $\mathbb{A}^3/\mu_r$ where $\mu_r$ acts by $\zeta \cdot (x,y,z) = (\zeta^a x, \zeta^b y, \zeta^c z)$ with $a + b + c \equiv 0 \pmod{r}$ and $\gcd(a, r) = \gcd(b, r) = \gcd(c, r) = 1$. The simplest is $\frac{1}{2}(1, 1, 1)$: $\mu_2$ acting by negation on $\mathbb{A}^3$.

Terminal singularities are precisely those quotient singularities that have no crepant partial resolution (except the identity). They play a key role in the minimal model program for threefolds.

If $G \subset \mathrm{Sp}(2n, k)$ acts on $\mathbb{A}^{2n}$ preserving the symplectic form, the quotient $\mathbb{A}^{2n}/G$ is a symplectic singularity. A resolution $Y \to \mathbb{A}^{2n}/G$ is symplectic if $Y$ carries a holomorphic symplectic form extending the one on the smooth locus.

For $n = 1$: $G \subset \mathrm{SL}(2) = \mathrm{Sp}(2)$, and the minimal resolution is always symplectic (=crepant). The ADE classification applies.

For $n = 2$: examples include wreath product quotients $(\mathbb{A}^2)^n / (G \wr S_n)$, whose symplectic resolutions are the Hilbert schemes $\operatorname{Hilb}^n(\tilde{S})$ where $\tilde{S} \to \mathbb{A}^2/G$ is the minimal resolution. This connects to Nakajima quiver varieties and the representation theory of affine Lie algebras.