G-groups of Cohen-Macaulay Rings with $n$-Cluster Tilting Objects

Let $(R,\mathfrak{m},k)$ denote a local Cohen-Macaulay ring such that the category of maximal Cohen-Macaulay $R$-modules $\mathbf{\text{mcm}}R$ contains an $n$-cluster tilting object $L$. In this paper, we compute $G_1® := K_1($\mathbf{\text{mod}}$\ R)$ explicitly as a direct sum of a free group and a specified quotient of ${\text{aut}}_R(L{)}_{\text{ab}}$ when $R$ is a $k$-algebra and $k$ is algebraically closed (and $\text{char}(k)\mathrm{\ne }2$). Moreover, we give some explicit computations of ${\text{aut}}_R(L{)}_{\text{ab}}$ and $G_1(R)$ for certain hypersurface singularities.