The Magic and Mystery of Component Tableaux

Abstract

Let $G$ be a simple algebraic group over the complex field $\mathbb{C}$, $P$ a parabolic subgroup containing $B$ its Borel subgroup, $P^{\prime }$ its derived group and $\mathfrak{m}$ the Lie algebra of its nilradical. The nilfibre $\mathcal{N}$ for this action is the zero locus of the augmentation ${\mathcal{I}}_{+}$ of the semi-invariant algebra $\mathcal{I}=\mathbb{C}[\mathfrak{m}{]}^{P^{\prime }}$. For $G=SL(n)$ practically nothing was known previously. The only result of comparable, but lesser complexity, is for $\mathcal{V}:=\mathcal{O}\cap \mathfrak{n}$, with $\mathcal{O}$ a nilptent $G$ orbit and $\mathfrak{n}$ the set of strictly upper triangular matrices. Then $\mathcal{V}$ is equidimensional with components known as orbital varieties, parameterised by standard tableaux whose shape is dictated by $\mathcal{O}$. Here the components of $\mathcal{N}$ are studied for $G=SL(n)$. They increase exponentially in $n$ with no a priori discernable pattern. For each choice of numerical data $\mathcal{C}$, a semi-standard tableau ${\mathcal{T}}^{\mathcal{C}}$, is constructed from $\mathcal{T}$. A \textit{delicate and tightly interlocking} analysis constructs a set of excluded root vectors from $\mathfrak{m}$ such that the complementary space ${\mathfrak{u}}^{\mathcal{C}}$ has the following properties. First it is a subalgebra of $\mathfrak{m}$. Secondly $\mathcal{C}:=\overline{B.{\mathfrak{u}}^{\mathcal{C}}}$ lies in $\mathcal{N}$ to which, thirdly, a Weierstrass section can be associated. Fourthly $\dim \mathcal{C}=dim\mathfrak{m}-\text{g}$, where \textbf{g} is the number of generators of the polynomial algebra $\mathcal{I}$. Fifthly the Weierstrass section, is shown to imply that $\mathcal{C}$ an irreducible component of $\mathcal{N}$, yet $\mathcal{C}$ is \textit{ only sometimes} an orbital variety closure. The resulting Component Map ${\mathcal{T}}^{\mathcal{C}}\mapsto \mathcal{C}$ is shown to be injective. Evidence for its surjectivity is given.