Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiplicities and dimensions in enveloping tensor categories

Friedrich Knop

TL;DR

The paper addresses the problem of determining tensor product multiplicities and internal dimensions of simple objects in the semisimple enveloping tensor category ${\mathcal T}({\mathcal A},\delta)$ built from a regular Mal'cev category ${\mathcal A}$. It develops subobject and subquotient decompositions as the central technical framework, enabling explicit multiplicity formulas via permutation characters and detailed dimension computations through Möbius inversion and fixed-point methods. A principal result is the general multiplicity formula for simple objects, together with a description of the Grothendieck ring and connections to stable Kronecker products in the ${\mathcal A}=\mathsf{Set}^{\mathrm{op}}$ case, and Deligne-type dimension expressions. The work thus provides concrete invariants for enveloping tensor categories and links to symmetric-function theory and potential interpolation categories, with broad implications for understanding semisimple tensor structures arising from finite algebraic theories.

Abstract

In the previous paper arxiv:math/0610552 semisimple tensor categories were constructed out of certain regular Mal'cev categories. In this paper, we calculate the tensor product multiplicities and the categorical dimensions of the simple objects. This yields also the Grothendieck ring. The main tool is the subquotient decomposition of the basic objects.

Multiplicities and dimensions in enveloping tensor categories

TL;DR

The paper addresses the problem of determining tensor product multiplicities and internal dimensions of simple objects in the semisimple enveloping tensor category built from a regular Mal'cev category . It develops subobject and subquotient decompositions as the central technical framework, enabling explicit multiplicity formulas via permutation characters and detailed dimension computations through Möbius inversion and fixed-point methods. A principal result is the general multiplicity formula for simple objects, together with a description of the Grothendieck ring and connections to stable Kronecker products in the case, and Deligne-type dimension expressions. The work thus provides concrete invariants for enveloping tensor categories and links to symmetric-function theory and potential interpolation categories, with broad implications for understanding semisimple tensor structures arising from finite algebraic theories.

Abstract

In the previous paper arxiv:math/0610552 semisimple tensor categories were constructed out of certain regular Mal'cev categories. In this paper, we calculate the tensor product multiplicities and the categorical dimensions of the simple objects. This yields also the Grothendieck ring. The main tool is the subquotient decomposition of the basic objects.
Paper Structure (10 sections, 28 theorems, 108 equations)

This paper contains 10 sections, 28 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Regular categories and their tensor envelopes
  3. The subobject decomposition
  4. The subquotient decomposition
  5. Tensor product multiplicities
  6. Dimensions
  7. The Möbius function of the lattice of quotient objects
  8. Symmetric function identities
  9. Proof of \ref{['thm:Littlewood']}
  10. Proof of \ref{['cor:identity']}

Key Result

Theorem 1.1

Let $[x_1]^0_{\chi_1}$ and $[x_2]^0_{\chi_2}$ be two simple objects of ${\mathcal{T}}$. Then

Theorems & Definitions (57)

  • Theorem 1.1: \ref{['cor:multtensor2']} below
  • Theorem 1.2: \ref{['cor:second']} below
  • Theorem 2.1: TERC*Thm. 6.1 and Thm. 8.3
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4.1: Subquotient decomposition
  • proof
  • ...and 47 more