Counting paths with Schur transitions

TL;DR

This paper introduces a new combinatorial formula for counting paths in the branching graph of the unitary group using Schur transitions, with implications for quantum generalizations and applications in RCFTs.

Contribution

It presents a novel combinatorial expression for path counting in the unitary group's branching graph and establishes its validity, hinting at a quantum generalization.

Findings

New combinatorial formula for path counting

Formal proof of the formula's validity

Introduction of quantum relative dimension

Abstract

In this work we explore the structure of the branching graph of the unitary group using Schur transitions. We find that these transitions suggest a new combinatorial expression for counting paths in the branching graph. This formula, which is valid for any rank of the unitary group, reproduces known asymptotic results. We proceed to establish the general validity of this expression by a formal proof. The form of this equation strongly hints towards a quantum generalization. Thus, we introduce a notion of quantum relative dimension and subject it to the appropriate consistency tests. This new quantity finds its natural environment in the context of RCFTs and fractional statistics; where the already established notion of quantum dimension has proven to be of great physical importance.