The unbroken spectrum of type-A Frobenius seaweeds

Abstract

If g is a Frobenius Lie algebra, then for certain F ∈ g * the natural map g −→ g * given by x −→ F [x, −] is an isomorphism. The inverse image of F under this isomorphism is called a principal element. We show that if g is a Frobenius seaweed subalgebra of An−1 = sl(n) then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicites have a symmetric distribution. Our proof methods are constructive and combinatorial in nature.Mathematics Subject Classification 2010 : 17B20, 05E15

The unbroken spectra of Frobenius seaweeds

“…But, in their formal study of principal elements [12], Gerstenhaber and Giaquinto showed that if g is a Frobenius seaweed subalgebra of A n−1 = sl(n), then the spectrum of the adjoint of a principal element of g consists entirely of integers. 1 Subsequently, the last three of the current authors showed that this spectrum must actually be an unbroken sequence of integers centered at one half [6]. 2 Moreover, the dimensions of the associated eigenspaces are shown to have a symmetric distribution.…”

“…Remark 4.11. Theorem 4.10 combined with the results of [6] gives us the following complete table of possible simple eigenvalues for each classical case.…”

“…Remark 1.2. In the prequel to this article [6], the type-A unbroken symmetric spectrum result is established using a combinatorial argument based on the graph-theoretic meander construction of Dergachev and Kirillov [8]. The combinatorial arguments heavily leverage the results of [6], but the inductions here are predicated on the basis-independent "orbit meander" construction of Joseph [16].…”

The Index and Spectrum of Lie Poset Algebras of Types B, C, and D

“…In [18], Ooms shows that the eigenvalues (and multiplicities) of ad( F ) = [ F , −] : g → g do not depend on the choice of principal element F . It follows that the spectrum of ad( F ) is an invariant of g, which we call the spectrum of g (see [1,3,7]).…”

Toral posets and the binary spectrum property

“…Seaweed (or biparabolic) subalgebras of a complex semi-simple Lie algebra g are intersections of two parabolic subalgebras whose sum is g (see[10,12]). It has been shown that the spectrum of seaweed subalgebras of the classical families of Lie algebras consists of an unbroken sequence of integers where the multiplicities of the eigenvalues form a symmetric distribution about one half (see[2,4]).…”