[Paper Review] An effective method to compute closure ordering for nilpotent orbits of $ heta$-representations

Willem A. de Graaf, É. B. Vinberg+1 more|arXiv (Cornell University)|Jul 10, 2011

Advanced Algebra and Geometry14 references3 citations

TL;DR

This paper presents an algorithm to compute the closure ordering of nilpotent orbits in θ-representations of simple complex Lie algebras, leveraging Hesselink strata and Weyl group actions to determine orbit inclusions via characteristic elements and orbit dimension comparisons. The method enables efficient computation of Hasse diagrams for nilpotent orbits in exceptional and classical θ-groups, with verified results matching known classifications for E7 and E8.

ABSTRACT

We develop an algorithm for computing the closure of a given nilpotent $G_0$-orbit in $\g_1$, where $\g_1$ and $G_0$ are coming from a $\Z$ or a $\Z/m\Z$-grading $\g= \bigoplus \g_i$ of a simple complex Lie algebra $\g$.

Motivation & Objective

• To develop an effective algorithm for determining whether one nilpotent G₀-orbit lies in the closure of another in θ-representations.
• To address the challenge of computing closure diagrams for nilpotent orbits in exceptional and classical θ-groups, where uniform classification is difficult.
• To provide a computational tool that replaces ad hoc case-by-case analysis with a systematic, algorithmic approach.
• To verify the correctness of the method by reproducing known Hasse diagrams for E7 and E8, and for real forms of exceptional Lie algebras.

Proposed method

• Use the Hesselink characteristic h of a nilpotent element e ∈ g₁ to define the Hesselink stratum V≥2(h), where G₀e = G₀(V≥2(h)).
• Determine orbit inclusion G₀e′ ⊂ G₀e by checking whether G₀e′ ∩ V≥2(h) is non-empty.
• Replace G₀ with a union of Bruhat cells to analyze orbit intersections effectively.
• For each w ∈ W₀ (the Weyl group of G₀), compute U(w) = V₂(h′) ∩ V≥2(wh), where V₂(h′) consists of vectors v with [h′, v] = 2v.
• Use dimension comparison: if dim Z(h′)v < dim Z(h′)e′ for all v ∈ V₂(h′) \\(G₀e′), then G₀e′ ∉ G₀e.
• Parametrize the Weyl group as a tree with edges from simple reflections to systematically explore orbit inclusions.

Experimental results

Research questions

• RQ1 How can one algorithmically determine whether a nilpotent G₀-orbit lies in the closure of another in a θ-representation?
• RQ2 What is the role of the Hesselink characteristic in characterizing orbit closures in θ-groups?
• RQ3 Can the closure ordering of nilpotent orbits in exceptional θ-groups be computed uniformly and verified computationally?
• RQ4 How do the closure diagrams of nilpotent orbits in complex and real forms of exceptional Lie algebras relate via the Kostant-Sekiguchi correspondence?
• RQ5 To what extent do degenerations of metabelian Lie algebras correspond to orbit closures in θ-representations?

Key findings

• The algorithm successfully reproduces the Hasse diagrams for nilpotent orbits in E₇ and E₈, matching the results of Mizuno and Beynon–Spaltenstein.
• The method correctly computes the closure diagrams for all θ-groups of order 2 in exceptional complex Lie algebras, verifying Djoković’s results.
• For the θ-group of type E₈ with signature (5,5), the variety of metabelian Lie algebras is irreducible and admits a one-parameter family of maximal GL(W)-orbits.
• For the θ-group of type E₈ with signature (6,3), the variety of metabelian Lie algebras is irreducible and admits a two-parameter family of maximal orbits.
• The lower parts of the Hasse diagrams for signatures (5,5) and (6,3) coincide, reflecting shared quotients L/Z with smaller signatures.
• The implementation in GAP correctly identifies orbit inclusions and produces consistent diagrams with labeled signatures using font distinctions as per Tables 2 and 3.