The Composition Tableau and Reconstruction of the Canonical Weierstrass Section for Parabolic Adjoint Action in type $A$

TL;DR

The paper introduces a composition tableau that simplifies the understanding of the canonical Weierstrass section for parabolic adjoint action in type A, providing a new, more straightforward proof and insights into the structure of the nilfibre.

Contribution

It constructs a composition tableau that recovers the canonical Weierstrass section without relying on previous complex references, simplifying the analysis of the nilfibre in type A.

Findings

The composition tableau accurately recovers the canonical Weierstrass section.

The approach simplifies the proof that $e+V$ is a Weierstrass section.

It enables reading off VS quadruplets, clarifying the structure of the canonical component.

Abstract

A "Composition map" is constructed, leaning heavily on earlier work [Y. Fittouhi and A. Joseph, Parabolic adjoint action, Weierstrass sections and components of the nilfibre in type $A$, Indag Math. and Y. Fittouhi and A. Joseph, The canonical component of the nilfibre for parabolic adjoint action, Weierstrass sections in type $A$, preprint, Weizmann, 2021]. It defines a composition tableau which recovers the "canonical" Weierstrass section $e+V$ described in the first paper above. Moreover \textit{without reference to this earlier work}, it is then shown that $e+V$ is indeed a Weierstrass section. This results in a huge simplification. Moreover one may read off from the composition tableau the "VS quadruplets'' of the second of the above papers, thereby describing the "canonical component" of the nil-fibre in which $e$ lies but does not of itself determine.