On the adjoint representation of $\mathfrak{sl}_n$ and the Fibonacci numbers

TL;DR

This paper introduces a combinatorial method to decompose the adjoint representation of re9le9 algebra re9le9 sl_{r+1} using Weyl groups, revealing a connection between the decomposition and Fibonacci numbers.

Contribution

It presents a novel combinatorial approach to decompose re9le9 representations and links the structure to Fibonacci numbers, providing new insights into Lie algebra representations.

Findings

The cardinality of the Weyl alternation set equals the Fibonacci number.

The method derives the exponents of re9le9 algebra from combinatorial properties.

A new connection between Lie algebra representation theory and Fibonacci numbers is established.

Abstract

We decompose the adjoint representation of $\mathfrak{sl}_{r+1}=\mathfrak {sl}_{r+1}(\mathbb C)$ by a purely combinatorial approach based on the introduction of a certain subset of the Weyl group called the \emph{Weyl alternation set} associated to a pair of dominant integral weights. The cardinality of the Weyl alternation set associated to the highest root and zero weight of $\mathfrak {sl}_{r+1}$ is given by the $r^{th}$ Fibonacci number. We then obtain the exponents of $\mathfrak {sl}_{r+1}$ from this point of view.