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Let $\mathbf{G}$ be either a simple linear algebraic group over analgebraically closed field of characteristic $\ell>0$ or a quantum group at an$\ell$-th root of unity. We define a tensor ideal of singular$\mathbf{G}$-modules in the category $\mathrm{Rep}(\mathbf{G})$ offinite-dimensional $\mathbf{G}$-modules and study the associated quotientcategory $\mathrm{\underline{Re}p}(\mathbf{G})$, called the regular quotient.Our main results are a 'linkage principle' and a 'translation principle' fortensor products: Let $\mathrm{\underline{Re}p}_0(\mathbf{G})$ be the essentialimage in $\mathrm{\underline{Re}p}(\mathbf{G})$ of the principal block of$\mathrm{Rep}(\mathbf{G})$. We first show that$\mathrm{\underline{Re}p}_0(\mathbf{G})$ is closed under tensor products in$\mathrm{\underline{Re}p}(\mathbf{G})$. Then we prove that the monoidalstructure of $\mathrm{\underline{Re}p}(\mathbf{G})$ is governed to a largeextent by the monoidal structure of $\mathrm{\underline{Re}p}_0(\mathbf{G})$.These results can be combined to give an external tensor product decomposition$\mathrm{\underline{Re}p}(\mathbf{G}) \cong \mathrm{Ver}(\mathbf{G}) \boxtimes\mathrm{\underline{Re}p}_0(\mathbf{G})$, where $\mathrm{Ver}(\mathbf{G})$denotes the Verlinde category of $\mathbf{G}$.

Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups