(The invariant ring)
Take \(\sigma = \Cone\big( m e_1 - e_2,\ e_2 \big) \subseteq N_\RR = \RR^2\) with \(m \geq 2\). Solving \(\inp{u}{m e_1 - e_2} \geq 0\) and \(\inp{u}{e_2} \geq 0\) gives \(\dualof{\sigma} = \Cone\big( (1,0), (1,m) \big)\), whose lattice points have Hilbert basis \((1,0), (1,1), \ldots, (1,m)\), so with \(x = \chi^{e_1}\) and \(y = \chi^{e_2}\), \[ A_\sigma = k[S_\sigma] = k\big[ x,\ xy,\ \ldots,\ xy^m \big] . \] Substituting \(x = u^m\) and \(y = v/u\) turns the generators into the degree-\(m\) monomials: \[ A_\sigma = k\big[ u^m,\ u^{m-1} v,\ \ldots,\ u v^{m-1},\ v^m \big] . \]
(The group)
Let \(\mu_m\) act diagonally on \(\AA^2\) by \[ \zeta \cdot (u,v) = (\zeta u,\ \zeta v), \qquad \zeta^m = 1 . \] A monomial \(u^i v^j\) is invariant exactly when \(i + j \equiv 0 \pmod m\), so the invariant ring is generated by the degree-\(m\) monomials, and \[ A_\sigma = k[u,v]^{\mu_m} , \qquad U_\sigma \cong \AA^2 / \mu_m . \] In the standard notation this is the cyclic quotient singularity \(\tfrac{1}{m}(1,1)\).
(The lattice reading of the same fact)
The two generators of \(\sigma\) span a sublattice of index \[ \abs{\det \begin{bmatrix} m & -1 \\ 0 & 1 \end{bmatrix}} = m \] inside \(N\), and that index is the order of the group. This is the general statement: a simplicial cone whose generators span a sublattice of index \(d\) gives the quotient of \(\AA^n\) by a group of order \(d\), and the group is \(N/N'\) for \(N'\) the sublattice. Thus the same determinant that detects non-smoothness gives the order of the finite quotient group.
For \(m = 2\) this is the quadric cone \(V(z_0 z_2 - z_1^2) = \AA^2/\pm 1\), the \(A_1\) surface singularity, and the same variety appears as the singular chart of \(\PP(1,1,2)\). The quotient description is the one that generalises: it survives to higher dimension and to non-diagonal weights, where the determinantal presentation by \(2 \times 2\) minors does not.