A characterization of the Delannoy category by Adams operations

2025/11/07 by Snowden, Andrew, Snyder, Noah · 1 citation
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2511.05768

openalex publication_date 2025/11/07 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

Abstract

In recent joint work with Harman, we studied a pre-Tannakian category called the Delannoy category, and showed that it had numerous special properties. One of these is that the Adams operations on its Grothendieck group are trivial. In this paper, we prove three theorems inspired by this. Theorem A states that the Delannoy category is the unique semi-simple pre-Tannakian category having a generator that is fixed by the second Adams operation and whose exterior powers are simple. Theorem B states the Delannoy category is uniquely determined by its Grothendieck semi-ring (among semi-simple pre-Tannakian categories). This is reminiscent of Kazhdan--Wenzl's recognition theorem for quantum \mathfrakgl_n, and many subsequent results. Finally, Theorem C establishes some properties of pre-Tannakian categories where the second Adams operation fixes a generator.

Citations

• Universal properties of Delannoy categories
• Reconstruction of tensor categories of type G₂
• Finite symmetric algebras in tensor categories and Verlinde categories of algebraic groups
• The thirty-seven measures on permutations
• Discrete pre-Tannakian categories
• The Delannoy category
• Remarks on symmetric fusion categories of low rank in positive characteristic
• Oligomorphic groups and tensor categories
• Classification of pivotal tensor categories with fusion rules related to SO(4)
• On Frobenius exact symmetric tensor categories
• Categories generated by a trivalent vertex

Cited by

• Classification of simple commutative algebras in the Delannoy category

Related

• The Delannoy category
• Classification of simple commutative algebras in the Delannoy category
• Tannakian categories: origins and summary
• A characterization of categories of coherent sheaves of certain algebraic stacks
• On the relationship between Galois and Tannakian categories: an open letter