The elegance of the component tableaux in type A

Let $G$ be a simple algebraic group over the complex field $\mathbb C$, $B$ a fixed Borel subgroup, $P$ a parabolic subgroup containing $B$, $P'$ 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 $\mathscr I:=\mathbb C[\mathfrak m]^{P'}$.  

In this discussion, we focus on the study of  $\mathscr N$ for $G=SL(n)$.  The composition of $n$ defined by the Levi block sizes in $P$ defines a standard tableau $\mathscr T$. For each choice of numerical data $\mathcal C$, a semi-standard tableau $\mathscr T^\mathcal C$, is constructed from $\mathscr T$. A delicate and tightly interlocking analysis constructs a set of excluded root vectors from $\mathfrak m$ such that the complementary space $\mathfrak u^\mathcal C$ is a subalgebra and a Weierstrass section can be associated to it. In addition, we will prove that  $\mathscr C:=\overline{B.\mathfrak u^\mathcal C}$ lies in $\mathscr N$ and its dimension is $\dim \mathfrak m-\textbf{g}$, where  \textbf{g} is the number of generators of the polynomial algebra $\mathscr I$.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062