A combinatorial approach to the index of seaweed subalgebras of Kac--Moody algebras
Oksana Yakimova
TL;DR
The paper develops a combinatorial, graph-based method to compute the index of seaweed subalgebras inside affine Kac–Moody algebras, focusing on types $\widetilde{\sf A}_r$ and $\widetilde{\sf C}_r$. It constructs affine meander graphs $\Gamma(S,S')$ and $\Gamma^{\sf C}(S,S')$ that encode the Kostant cascade data and parabolic choices, and then derives explicit index formulas via the TYJ framework: for type $\widetilde{\sf A}_r$, $\mathsf{ind\,}\bar{\mathfrak q}=|\text{segments}|+2|\text{cycles}|-\iota$ with $\iota\in\{0,2\}$; for type $\widetilde{\sf C}_r$, $\mathsf{ind\,}\bar{\mathfrak q}=1+|\text{cycles}|+\tfrac{1}{2}|\text{non-$\sigma$-stable segments}|-\iota$ with a similar $\iota$-dependence. The TYJ formula then justifies these combinatorial expressions and connects them to Kostant cascades and orbit counts. The work also revisits the finite-dimensional and orthogonal cases, showing that the same TYJ-based approach yields known index formulas via corresponding meander-type graphs. Overall, the results provide a practical, diagrammatic toolkit for index computation in affine seaweed subalgebras of type $\mathsf{A}$ and $\mathsf{C}$, with extensions to the orthogonal family discussed.
Abstract
In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in $\mathfrak{gl}_n$ and computed their index using certain graphs. Then seaweed subalgebras $\mathfrak q\subset\mathfrak g$ were defined by Panyushev for any reductive $\mathfrak g$. A few years later Joseph generalised this notion to the setting of (untwisted) affine Kac--Moody algebras $\widehat{\mathfrak g}$. Furthermore, he proved that the index of such a seaweed can be computed by the same formula that had been known for $\mathfrak g$. In this paper, we construct graphs that help to understand the index of a seaweed $\mathfrak q\subset\widehat{\mathfrak g}$, where $\widehat{\mathfrak g}$ is of affine type A or C.