# Cauchy formula and the character ring

## TL;DR

This paper explores the fundamental role of the Cauchy summation formula in character calculus, demonstrating its equivalence to expressing skew characters via Littlewood-Richardson coefficients, with a detailed example involving plane partitions.

## Contribution

It clarifies the relationship between the Cauchy formula and skew characters, providing a specific example with plane partitions at a minimal non-trivial level.

## Key findings

- Cauchy formula is equivalent to expressing skew characters through Littlewood-Richardson coefficients.
- Illustration with plane partitions at four boxes demonstrates the equivalence.
- Highlights the importance of the Cauchy formula in character calculus applications.

## Abstract

Cauchy summation formula plays a central role in application of character calculus to many problems, from AGT-implied Nekrasov decomposition of conformal blocks to topological-vertex decompositions of link invariants. We briefly review the equivalence between Cauchy formula and expressibility of skew characters through the Littlewood-Richardson coefficients. As not-quite-a-trivial illustration we consider how this equivalence works in the case of plane partitions -- at the simplest truly interesting level of just four boxes.