The Magic and Mystery of Component Tableaux

TL;DR

This work advances the understanding of the nilfibre \mathscr N for the parabolic adjoint action in type A by introducing component tableaux, a novel combinatorial framework that parametrizes irreducible components via subalgebras \mathfrak{u}^{\\mathcal C} of the nilradical. Each component tableau yields a Weierstrass section and a corresponding component \mathscr C = \overline{B.\\mathfrak u^{\\mathcal C}} in \mathscr N, with a proven injective component map; surjectivity is demonstrated in parts and pursued via a broader program. The approach circumvents the inadequacy of Bruhat-based methods, connects to Benlolo–Sanderson invariants, and situates the results within the broader study of invariant theory, orbital varieties, and possible quantization. The findings illuminate when components are orbital-variety closures and when they are not, revealing a rich and intricate combinatorial structure that grows rapidly with n and invites further exploration of surjectivity and generalization beyond type A.

Abstract

Let $G$ be a simple algebraic group over the complex field $\mathbb C$, $P$ a parabolic subgroup containing $B$ its Borel subgroup, $P'$ its derived group and $\mathfrak m$ the Lie algebra of its nilradical. The nilfibre $\mathscr N$ for this action is the zero locus of the augmentation $\mathscr I_+$ of the semi-invariant algebra $\mathscr I=\mathbb C[\mathfrak m]^{P'}$. For $G=SL(n)$ practically nothing was known previously. The only result of comparable, but lesser complexity, is for $\mathscr V:=\mathscr O\cap \mathfrak n$, with $\mathscr O$ a nilptent $G$ orbit and $\mathfrak n$ the set of strictly upper triangular matrices. Then $\mathscr V$ is equidimensional with components known as orbital varieties, parameterised by standard tableaux whose shape is dictated by $\mathscr O$. Here the components of $\mathscr N$ 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 $\mathscr T^\mathcal C$, is constructed from $\mathscr T$. 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 $\mathscr C:=\overline{B.\mathfrak u^\mathcal C}$ lies in $\mathscr N$ to which, thirdly, a Weierstrass section can be associated. Fourthly $\dim \mathscr C = dim \mathfrak m-\textbf{g}$, where \textbf{g} is the number of generators of the polynomial algebra $\mathscr I$. Fifthly the Weierstrass section, is shown to imply that $\mathscr C$ an irreducible component of $\mathscr N$, yet $\mathscr C$ is \textit{ only sometimes} an orbital variety closure. The resulting Component Map $\mathscr T^\mathcal C\mapsto\mathscr C$ is shown to be injective. Evidence for its surjectivity is given.

Figures (15)

• Figure 1: This illustrates Overview, \ref{['3.2.6']}. The dashed green lines are neutral lines from $C_r(\infty)$ to $C_{r+1}(\infty)$. The solid black lines with label $1$ join an entry in $C_r(\infty)$ to an entry in $C_{r+1}$. This is also true of the dashed/dotted blue lines, but the existence of the latter is conditional on their being an entry in the left hand factor and from which there is no outgoing neutral line. (In particular the uppermost dashed/dotted line would be absent if there were no entry $i_{m_1+m+2+1}$ in the left hand column or if it were joined to an entry in the right hand column by a neutral line, as in top diagram rather the bottom diagram in Fig. $3$.) The vertical red lines join an entry of $C_{r+1}(\infty)\setminus C_{r+1}$ to $C_{r+1}$.
• Figure 2: Consider the composition $(1,2,1,2)$. There are two component tableaux. Starting from $\mathscr T$, one first moves $2$ below $4$. Then $\mathscr B^2_{2,4} = \{3,2\}$. Thus either $2$ can be lowered below $6$ or $3$ below $2$ and translated horizontally under $6$.
• Figure 3: This describes the decoration of the lines in $\mathscr T(\infty)$ coming from the two component tableaux of Figure $2$ following \ref{['3.2.6']} and Figure $1$. Notice that in the top diagram, there is no line (with label $1$) to $6$ because there is no entry in the box $R_3\cap C_3$ (which could be remedied by adding a column of height $3$ on the extreme left) and because $2$ is joined by a neutral line to $C_3$.
• Figure 4: This describes the three component tableaux for the composition $(2,1,1,2,1)$ following assiduously the rules of section \ref{['3.2.2']}. Notably in the bottom tableau $4$ is lowered by two rows below $6$ using the fact that $4 \in \mathscr B^1_{3,5} \cap \mathscr B^2_{1,4}$ and this excludes any further use of these pairs of neighbouring columns.
• Figure 5: This describes the decoration of the lines in Figure $4$.

Theorems & Definitions (72)

• Lemma
• proof
• Lemma
• proof
• Corollary
• proof
• Lemma
• proof
• Lemma
• proof