Multiparameter colored partition category and the product of the reduced Kronecker coefficients

We introduce and study a multiparameter colored partition category $\mathcal{C}\mathcal{P}\mathcal{a}\mathcal{r}(\mathbf{\text{x}})$ by extending the construction of the partition category, over an algebraically closed field $\mathbb{k}$ of characteristic zero and for a multiparameter $\mathbf{\text{x}}\in {\mathbb{k}}^r$. The morphism spaces in $\mathcal{C}\mathcal{P}\mathcal{a}\mathcal{r}(\mathbf{\text{x}})$ have bases in terms of partition diagrams whose parts are colored by elements of the multiplicative cyclic group $C_r$. We show that the endomorphism spaces of $\mathcal{C}\mathcal{P}\mathcal{a}\mathcal{r}(\mathbf{\text{x}})$ and additive Karoubi envelope of $\mathcal{C}\mathcal{P}\mathcal{a}\mathcal{r}(\mathbf{\text{x}})$ are generically semisimple. The category $\mathcal{C}\mathcal{P}\mathcal{a}\mathcal{r}(\mathbf{\text{x}})$ is rigid symmetric strict monoidal and we give a presentation of $\mathcal{C}\mathcal{P}\mathcal{a}\mathcal{r}(\mathbf{\text{x}})$ as a monoidal category. The path algebra of $\mathcal{C}\mathcal{P}\mathcal{a}\mathcal{r}(\mathbf{\text{x}})$ admits a triangular decomposition with Cartan subalgebra being equal to the direct sum of the group algebras of complex reflection groups $G(r,n)$. We compute the structure constants for the classes of simple modules in the split Grothendieck ring of the category of modules over the path algebra of the downward partition subcategory of $\mathcal{C}\mathcal{P}\mathcal{a}\mathcal{r}(\mathbf{\text{x}})$ in two ways. Among other things, this gives a closed formula for the product of the reduced Kronecker coefficients in terms of the Littlewood–Richardson coefficients for $G(r,n)$ and certain Kronecker coefficients for the wreath product $(C_r \times C_r)\wr S_n$\mathrm{.}For$r=1$, this formula reduces to a formula for the reduced Kronecker coefficients given by Littlewood. We also give two analogues of the Robinson–Schensted correspondence for colored partition diagrams and, as an application, we classify the equivalence classes of Green’s left, right and two-sided relations for the colored partition monoid in terms of these correspondences.