On seaweed subalgebras and meander graphs in type C

Abstract

In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in $\mathfrak{gl}(n)$ and computed their index using certain graphs. In this article, those graphs are called type-A meander graphs. Then the subalgebras of seaweed type, or just "seaweeds", have been defined by Panyushev (2001) for arbitrary simple Lie algebras. Namely, if $\mathfrak p_1,\mathfrak p_2\subset\mathfrak g$ are parabolic subalgebras such that $\mathfrak p_1+\mathfrak p_2=\mathfrak g$, then $\mathfrak q=\mathfrak p_1\cap\mathfrak p_2$ is a seaweed in $\mathfrak g$. A general algebraic formula for the index of seaweeds has been proposed by Tauvel and Yu (2004) and then proved by Joseph (2006). If $\mathfrak p_1$ and $\mathfrak p_2$ are "adapted" to a fixed triangular decomposition of $\mathfrak g$, then $\mathfrak q$ is said to be standard. The number of standard seaweeds is finite. In this paper, elaborating on the "graphical" approach of Dergachev and Kirillov, we introduce the type-C meander graphs, i.e., the graphs associated with the standard seaweeds of $\mathfrak{sp}(2n)$, and give a formula for the index in terms of these graphs. We also note that the very same graphs can be used in case of the odd orthogonal Lie algebras. We also provide several applications of our formula to the Frobenius seaweeds in $\mathfrak{sp}(2n)$. In particular, using a natural partition of the set $\mathcal F_n$ of standard Frobenius seaweeds, we prove that $\# \mathcal F_n$ strictly increases for the passage from $n$ to $n+1$. The similar monotonicity question is open for the standard Frobenius seaweeds in $\mathfrak{sl}(n)$, even for the passage from $n$ to $n+2$.