Reverse Tableaux and the Surjectivity of the Component Map in Type $A$

Abstract

Let $G = \mathrm{SL}(n,\mathbb{C})$, let $B$ be a fixed Borel subgroup, and let $P \supset B$ be a parabolic subgroup determined by a composition $(c_1,\dots,c_k)$ of $n$. Write $P'$ for the derived group of $P$ and $\mathfrak{m}$ for the Lie algebra of the nilradical of $P$. By Richardson's theorem the algebra of semi-invariants $\mathscr{I} := \mathbb{C}[\mathfrak{m}]^{P'}$ is polynomial; in type $A$ its generators may be taken to be the Benlolo--Sanderson (BS) invariants. The \emph{nilfibre} is the common zero locus $\mathscr{N} := V(\mathscr{I}_{+}) \subset \mathfrak{m}$. A set of \emph{component tableaux}, each encoding combinatorial data summarised in a multi-set called the \emph{Red Set}, was constructed in earlier work by Y. Fittouhi and A. Joseph in The reverse tableau: a gateway to the surjectivity of the component map. The resulting \emph{component map} $\phi : \{\text{component tableaux}\} \to \Irr(\mathscr{N})$ was shown to be injective. In the present article, we develop the Factorization Principle for Benlolo--Sanderson invariants in order to give a rigorous proof of the surjectivity of the component map $\phi$. While the combinatorial framework of reverse tableaux was introduced in a work by Y. Fittouhi and A. Joseph cited above, the surjectivity of $\phi$ remained conjectural: the linearization method used there did not exclude the possible loss or merging of irreducible components. The present paper resolves this geometric difficulty by showing that the relevant invariants factorize into products indexed by pseudo-neighbouring column pairs, thereby ensuring that every component is reached in a controlled and accountable way.