A characterization of the Delannoy category by Adams operations

TL;DR

This paper characterizes the Delannoy category as a unique semi-simple pre-Tannakian category with specific properties related to Adams operations and its Grothendieck semi-ring, extending understanding of its structure.

Contribution

It proves the uniqueness of the Delannoy category based on Adams operations and its Grothendieck semi-ring, providing new recognition theorems for such categories.

Findings

Delannoy category is uniquely determined by its properties.

Adams operations are trivial on the Grothendieck group of the Delannoy category.

The second Adams operation fixes generators in certain pre-Tannakian categories.

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 $\mathfrak{gl}_n$, and many subsequent results. Finally, Theorem C establishes some properties of pre-Tannakian categories where the second Adams operation fixes a generator.