Indecomposable Objects of $\underline{Rep}(GL_t)$ in Terms of Exterior Powers of the Tautological Object and its Dual

TL;DR

This paper provides a formula for the classes of indecomposable objects in the Grothendieck ring of the Deligne category Rep(GL_t), expressing them via exterior powers of the tautological object and its dual, enhancing understanding of the category's structure.

Contribution

It introduces a new formula for indecomposable objects in Rep(GL_t) using exterior powers, clarifying their algebraic structure.

Findings

Derived explicit formulas for indecomposable objects.

Connected the Grothendieck ring elements with exterior powers.

Enhanced understanding of the Deligne category Rep(GL_t) structure.

Abstract

The purpose of this short note is to understand the Grothendieck ring of the Deligne category $\underline{\text{Rep}}(GL_t)$. We give a formula for the class of an indecomposable object in terms of exterior powers of the tautological object and its dual.