On compositions associated to Frobenius parabolic and seaweed subalgebras of $\mathrm{sl}_{n}(\Bbbk )$

TL;DR

This paper establishes a bijective correspondence between Frobenius parabolic and seaweed subalgebras of sl_n and certain monoid elements, proving a conjecture on their enumeration based on compositions.

Contribution

It introduces a novel monoid-based framework to classify Frobenius subalgebras and proves a conjecture on their count related to compositions and pairs of compositions.

Findings

Bijection between subalgebras and monoid elements

Proof of a conjecture on the number of subalgebras

Enumeration formula based on compositions

Abstract

By using a free monoid of operators on the set of compositions (resp. pairs of compositions), we establish in this paper a bijective correspondence between Frobenius standard parabolic (resp. seaweed) subalgebras and certain elements of this monoid. We prove via this correspondence a conjecture of one of the authors on the number of Frobenius standard parabolic (resp. seaweed) subalgebras of $\mathrm{sl}_{n}(\Bbbk )$ associated to compositions (resp. pairs of compositions) with $n-t$ parts (resp. parts in total).