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

Gruber J (2024)

**Publication Type:** Journal article

**Publication year:** 2024

### Journal

**Book Volume:** 129

**Article Number:** e12641

**Journal Issue:** 4

**DOI:** 10.1112/plms.12641

### Abstract

Let (Formula presented.) be either a simple linear algebraic group over an algebraically closed field of characteristic (Formula presented.) or a quantum group at an (Formula presented.) -th root of unity. We define a tensor ideal of singular (Formula presented.) -modules in the category (Formula presented.) of finite-dimensional (Formula presented.) -modules and study the associated quotient category (Formula presented.), called the regular quotient. For (Formula presented.), the Coxeter number of (Formula presented.), we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let (Formula presented.) be the essential image in (Formula presented.) of the principal block of (Formula presented.). We first show that (Formula presented.) is closed under tensor products in (Formula presented.). Then we prove that the monoidal structure of (Formula presented.) is governed to a large extent by the monoidal structure of (Formula presented.). These results can be combined to give an external tensor product decomposition (Formula presented.), where (Formula presented.) denotes the Verlinde category of (Formula presented.).

### Authors with CRIS profile

### How to cite

**APA:**

Gruber, J. (2024). Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups. *Proceedings of the London Mathematical Society*, *129*(4). https://doi.org/10.1112/plms.12641

**MLA:**

Gruber, Jonathan. "Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups." *Proceedings of the London Mathematical Society* 129.4 (2024).

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