# Multiplicity formulas for fundamental strings of representations of classical Lie algebras

## TL;DR

This paper derives explicit formulas for the weight multiplicities of a specific class of irreducible representations, called p-fundamental strings, of classical Lie algebras, enhancing understanding of their structure.

## Contribution

It provides the first closed-form explicit formulas for weight multiplicities in p-fundamental strings of classical Lie algebras.

## Key findings

- Closed formulas for weight multiplicities in p-fundamental strings.
- Applicable to all classical Lie algebras and any p.
- Advances representation theory by explicit multiplicity calculations.

## Abstract

We call the \emph{$p$-fundamental string} of a complex simple Lie algebra to the sequence of irreducible representations having highest weights of the form $k\omega_1+\omega_p$ for $k\geq0$, where $\omega_j$ denotes the $j$-th fundamental weight of the associated root system. For a classical complex Lie algebra, we establish a closed explicit formula for the weight multiplicities of any representation in any $p$-fundamental string.