English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We introduce and study a multiparameter colored partition category$\mathcal{CPar}(\textbf{x})$ by extending the construction of the partitioncategory, over an algebraically closed field $\Bbbk$ of characteristic zero andfor a multiparameter $\textbf{x}\in \Bbbk^{r}$. The morphism spaces in$\mathcal{CPar}(\textbf{x})$ have bases in terms of partition diagrams whoseparts are colored by elements of the multiplicative cyclic group $C_r$. We showthat the endomorphism spaces of $\mathcal{CPar}(\textbf{x})$ and additiveKaroubi envelope of $\mathcal{CPar}(\textbf{x})$ are generically semisimple.The category $\mathcal{CPar}(\textbf{x})$ is rigid symmetric strict monoidaland we give a presentation of $\mathcal{CPar}(\textbf{x})$ as a monoidalcategory. The path algebra of $\mathcal{CPar}(\textbf{x})$ admits a triangulardecomposition with Cartan subalgebra being equal to the direct sum of the groupalgebras of complex reflection groups $G(r,n)$. We compute the structureconstants for the classes of simple modules in the split Grothendieck ring ofthe category of modules over the path algebra of the downward partitionsubcategory of $\mathcal{CPar}(\textbf{x})$ in two ways. Among other things,this gives a closed formula for the product of the reduced Kroneckercoefficients in terms of the Littlewood--Richardson coefficients for $G(r,n)$and certain Kronecker coefficients for the wreath product $(C_r \times C_r)\wrS_n$. For $r=1$, this formula reduces to a formula for the reduced Kroneckercoefficients given by Littlewood. We also give two analogues of theRobinson--Schensted correspondence for colored partition diagrams and, as anapplication, we classify the equivalence classes of Green's left, right andtwo-sided relations for the colored partition monoid in terms of thesecorrespondences.

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