In this paper we give a classification, for all the simple $L=A_r$ and $L=B_r$ of all the $\pi$ with $r(\pi)< r$.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:Given a semisimple (preferably simple) complex Lie algebra $L$, we consider the monoid $\Gamma=\Gamma(L)$ of equivalence classes of the finite dimensional reducible complex representations of $L$.
📊
Key Findings & Data:Here $\Gamma$ is identified with the lattice of the corresponding highest weights. (This equips $\Gamma$ with the monoid structure.) For $\pi\in\Gamma$ one considers the symmetric algebra $\displaystyle S(\pi)=\bigoplus_{n=0}^{\infty}S^n(\pi)$ (here regarded as a representation).
💡
Conclusion & Impact:In this paper we give a classification, for all the simple $L=A_r$ and $L=B_r$ of all the $\pi$ with $r(\pi)< r$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Lie algebra · Lie group · reducible representation
Abstract
Given a semisimple (preferably simple) complex Lie algebra $L$, we consider the monoid $\Gamma=\Gamma(L)$ of equivalence classes of the finite dimensional reducible complex representations of $L$. Here $\Gamma$ is identified with the lattice of the corresponding highest weights. (This equips $\Gamma$ with the monoid structure.) For $\pi\in\Gamma$ one considers the symmetric algebra $\displaystyle S(\pi)=\bigoplus_{n=0}^{\infty}S^n(\pi)$ (here regarded as a representation). The elements of $\Gamma$ ``occurring'' in $S(\pi)$ -- i.e., which are the highest weights of some irreducible component of the representation $S(\pi)$ -- form a subsemigroup $M(\pi)$ of $\Gamma$. Such a $M(\pi)$ has a naturally defined rank $r(\pi)$ with $1\leq r(\pi)\leq r = \text{rank of }L$. In this paper we give a classification, for all the simple $L=A_r$ and $L=B_r$ of all the $\pi$ with $r(\pi)< r$.
GÜNER, R. (2022). Reducible good representations of semisimple Lie algebras $A_r$ and $B_r$ Part I. Hacettepe Journal of Mathematics and Statistics, Cilt: 48 - Sayı: 4, 1146–1155. https://ajindex.com/articles/529789
A.G. Elashvili, Canonical form and stationary subalgebras of points of general position for simple linear Lie groups, Funct. Anal. Appl. 6, 44–53, 1972
R. Güner, Klassifikation gewisser Darstellungen halbeinfacher Liealgebren,
W.-Chung Hsiang and W.-Yi Hsiang, Differential actions of compact connected classical groups: II, Ann. of Math. (2), 91–92, 1970
J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer- Verlag, New York-Heidelberg-Berlin, 1972
M. Krämer, Hauptisotropiegruppen bei endlichen dimensionalen Darstellungen kompakter halbeinfacher Liegruppen, Diplomarbeit, Bonn, 1966
M. Krämer, Über Untergruppen kompakter Liegruppen als Isotropiegruppen bei linearen Aktionen, Math. Z. 147, 207–224, 1976
D.I. Panyushev, Complexity and rank of homegeneous spaces, Geom. Dedicata, 34, 249–269, 1990
J. Tits, Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, Lecture Notes in Mathematics No. 40, Berlin-Heidelberg-New York, Springer, 1967
M.A.A. Van Leeuwen, A.M. Cohen and B. Lisser, A package for Lie group computations, Can., Amsterdam 1992.