The Reverse Tableaux: a Gateway to the Surjectivity of the Component Map

Abstract

Let $G$ be a simple algebraic group over $\mathbb C$, $B$ a fixed Borel subgroup, $P$ a parabolic subgroup, $P'$ its derived group acting on the Lie algebra $\mathfrak m$ of its nilradical. The nilfibre $\mathscr N$ is the zero locus of the augmentation $\mathcal I_+$ of the semiinvariant algebra $\mathcal I=\mathbb C[\mathfrak m]^{P'}$. Via Richardson's theorem, $\mathcal I$ is polynomial. Then the generators of $\mathcal I$ may be taken to be the Benlolo-Sanderson invariants \cite{BS}. In Y.Fittouhi and A.Joseph, The Magic and Mystery of Component Tableaux, Indag 2026, a set $\{\mathscr T^\mathcal C\}$ of component tableaux was constructed each encoding explicit combinatorial data $\mathcal C$. Each tableau $\{\mathscr T^\mathcal C\}$ defines a component $\mathscr C$ of $\mathscr N$ and the map $\{\mathscr T^\mathcal C\}\mapsto \mathscr C$ is injective. Here this data is simply encoded in a multiset called the Red Set. In the present work a set $\{\mathscr R^{\psi(\mathcal C)}\}$ of Reverse Tableaux is constructed through an Enabling Proposition. They define the same components as the component tableaux and furthermore give a factorisation of each new invariant in a chosen sequence via successive linearisation of preceding invariants. Via Krull's theorem this factorisation provides the required surjectivity. There can be several reverse tableaux for a given RedSet, yet each determine the same variety as the component tableaux with the given RedSet and define the same components. The flexibility of having several equivalent reverse tableaux absent from the more rigid component tableaux, is essential for factorisation This procedure is different from the classical approach to surjectivity requiring a \textit{geometric} description of the nilcone which seems unattainable. Nothing of this complexity has never been tackled before and the methods used here are entirely new