Stability of Multiplicities in Symmetry Breaking: The sl_2 Case

TL;DR

This paper develops a universal, inequality-based framework for understanding how multiplicities in representation branching laws depend on parameters, exemplified through the sl_2 case, with broad applications.

Contribution

It introduces the notion of fences and a piecewise-linear approach to describe multiplicity stability, unifying classical phenomena in representation theory.

Findings

Multiplicities are governed by linear inequalities and fences.

Explicit formulas for stability and multiplicity variation are provided.

The framework applies to both finite-dimensional and admissible smooth representations.

Abstract

This expository paper explains, in the case of $\mathfrak{sl}_2$, the ideas introduced in the preprints (arXiv:2509.17007, 2604.22262), which develop a new framework for the study of multiplicities in branching laws of representations, with particular emphasis on their dependence on representation parameters. Taking the Lie algebra $\mathfrak{sl}_2$ as a guiding example, we show that multiplicities, which are often computed via ad hoc, case-by-case arguments, are in fact governed by universal systems of linear inequalities. To describe these inequalities, we introduce the notion of \emph{fences}, which encode the piecewise-linear boundaries of regions in parameter space on which multiplicities remain constant. Within this framework, we give an explicit description of how multiplicities vary as parameters move inside reduced coherent families of representations. Our approach applies uniformly both to finite-dimensional representations and to admissible smooth Fr\'echet representations of real reductive Lie groups, and reveals a subtle and intrinsic interplay between the parameters of a group and those of its subgroup. As an application of the general theory, we establish stability results and explicit formulas that clarify and unify a variety of classical phenomena, including the Pieri rule, $K$-type formulas, fusion rules, and tensor products of Verma modules. In particular, the stability of fusion multiplicities provides a concrete manifestation of the theory. More broadly, this framework suggests a unified approach to branching multiplicities extending beyond the $\mathfrak{sl}_2$ case.