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arXiv · 2211.06851

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

Yasmine Fittouhi·Anthony Joseph
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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.

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BibTeXRIS

Yasmine Fittouhi, Anthony Joseph. 2022-11-13. The Composition Tableau and Reconstruction of the Canonical Weierstrass Section for Parabolic Adjoint Action in type $A$. https://arxiv.org/abs/2211.06851

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