Symplectic meanders

Abstract

Analogous to the sl(n) case, we address the computation of the index of seaweed subalgebras of sp(2n) by introducing graphical representations called symplectic meanders. Formulas for the algebra's index may be computed by counting the connected components of its associated meander. In certain cases, formulas for the index can be given in terms of elementary functions. Mathematics Subject Classification 2010 : 17B08, 17B20The results of this paper were delivered by the second author in a talk entitled Symplectic Meanders in a January 2015 conference at the University of Miami in honor of his adviser Michelle Wachs. Some of these results, inclusive of the meander construction, have been independently obtained by D. Panyushev and O. Yakimova per a recent arXiv post on January 3, 2016 [12].

Meander Graphs and Frobenius Seaweed Lie Algebras III

“…Such combinatorial formulas have been established in the Type-A case by Dergachev and A. Kirillov [4] and in the Type-C case by Coll et al [11] (and independently by Panyushev and Yakimova [5]). These elegant formulas are difficult to apply in practice, but in certain cases can be replaced by closed form linear greatest common divisor formulas based on the flags that define the seaweed in its standard representation.…”

“…It is therefore not a priori clear to what extent the homotopy type is an algebraic invariant. However, the following theorem follows from Theorem 5.3 in the recent paper by Moreau and Yakimova [11]. Theorem 3.6: Conjugate seaweed subalgebras of sl(n) have the same homotopy type.…”

“…In fact, this remains an open question for an arbitrary seaweed. However, it is implicit in the work of Moreau and Yakimova [11] that for Type-A seaweeds the homotopy type is a conjugation invariant which is more granular than the index. The example at the end of this paper provides two seaweed algebras which have the same dimension, rank, and index − but different homotopy types.…”

The unbroken spectrum of type-A Frobenius seaweeds

“…We let M S denote the set of indices (i, j) such that there is a directed edge from i to j in M . (See the right side of Figure 2 where the directed edges of M are {(2, 1), (6,3), (5,4), (2,3), (4, 6)} and F MS = e * 2,1 + e * 6,3 + e * 5,4 + e * 2,3 + e * 4,6 .) The Dergachev-Kirillov functional is regular in the sense that F MS realizes the smallest possible dimension for the kernel of F ([y, −]) where F ranges over all linear functionals.…”

“…Moreover, the dimensions of the associated eigenspaces form a symmetric distribution about one half. 6 The proof of Theorem 3…”