The Magic and Mystery of Component Tableaux
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arXiv
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| Formato: | Preprint |
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2024
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| _version_ | 1866909894431473664 |
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| author | Fittouhi, Yasmine Joseph, Anthony |
| author_facet | Fittouhi, Yasmine Joseph, Anthony |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05935 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Magic and Mystery of Component Tableaux Fittouhi, Yasmine Joseph, Anthony Representation Theory 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. |
| title | The Magic and Mystery of Component Tableaux |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2407.05935 |