Series of Lie groups

By J. M. Landsberg, Laurent Manivel. 2002

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For various series of complex semi-simple Lie algebras $\fg (t)$ equipped with irreducible representations $V(t)$, we decompose the tensor powers of $V(t)$ into irreducible factors in a uniform manner, using a tool we call {\it diagram induction}. In particular, we interpret the decompostion formulas of Deligne \cite{del} and Vogel \cite{vog} for decomposing $\fg^{\ot k}$ respectively for the exceptional series and $k\leq 4$ and all simple Lie algebras and $k\leq 3$, as well as new formulas for the other rows of Freudenthal's magic chart. By working with Lie algebras augmented by the symmetry group of a marked Dynkin diagram, we are able to extend the list \cite{brion} of modules for which the algebra of invariant regular functions under a maximal nilpotent subalgebra is a polynomial algebra. Diagram induction applied to the exterior algebra furnishes new examples of distinct representations having the same Casimir eigenvalue.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c9ec70871a5fe085f2df1ba365d12c360a5f01df

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Citations in the corpus

Ranks of tensors and a generalization of secant varieties (2009, background): “Proposition 4.1. Let X ⊂ PV be one of v2(Pn) (the rank one symmetric (n + 1) × (n + 1) matrices), G(2,n) (the rank two skew-symmetric n × n matrices), Seg(Pa × Pb), the Cayley plane OP2, or the 10-dimensional spinor variety S5, (i.e, X is a sub-cominuscule variety, see [ 17 ]).”