# Combinatorial index formulas for Lie algebras of seaweed type

## TL;DR

This paper develops combinatorial formulas to compute the index of seaweed subalgebras in Lie algebras of type D, extending previous types, and provides closed-form polynomial gcd formulas based on associated meander structures.

## Contribution

It introduces comprehensive combinatorial formulas for the index of seaweed subalgebras in type D Lie algebras using meander tail analysis, extending classical types.

## Key findings

- Formulas for the index based on connected components of meanders.
- Closed-form polynomial gcd formulas for the index.
- Unified combinatorial approach for classical types.

## Abstract

Analogous to the types A, B, and C cases, we address the computation of the index of seaweed subalgebras in the type-D case. Formulas for the algebra's index can be computed by counting the connected components of its associated meander. We focus on a set of distinguished vertices of the meander, called the tail of the meander, and using the tail, we provide comprehensive combinatorial formulas for the index of a seaweed in all the classical types. Using these formulas, we provide all general closed-form index formulas where the index is given by a polynomial greatest common divisor formula in the sizes of the parts that define the seaweed.