Degeneration order of $3\times 3$ nilpotent matrix tuples

Abstract: The degeneration order of simultaneous similarity classes of $3\times 3$ nilpotent matrix tuples is determined, and is shown to be given by rank conditions.

• The paper proves that for 3x3 nilpotent matrix tuples, orbit degeneration and hom-order are equivalent under rank conditions.
• It employs a module-theoretic framework and explicit orbit classification to map degeneration orders via rank inequalities.
• The results bridge geometric invariant theory and representation theory, providing a computable method for orbit closure analysis.

Degeneration Order of $3\times 3$ Nilpotent Matrix Tuples

Introduction and Motivation

The paper "Degeneration order of $3\times 3$ nilpotent matrix tuples" (2604.22609) investigates the relationship between orbit closures and degeneration orders for tuples of nilpotent $3\times 3$ matrices under simultaneous conjugation by $GL_3(K)$, where $K$ is an algebraically closed field. Studying the degeneration order is motivated by geometric complexity theory and invariant theory, which often require determining whether the orbit of one tuple lies within the closure of another with respect to the Zariski topology. Given the wildness of the classification problem for similarity classes in $(K^{n\times n})^2$ for general $n$, the focus on nilpotent $3\times 3$ matrix tuples makes the analysis tractable and provides a comprehensive characterization.

Preorders and Orbit Closures

Two distinct preorders are considered on tuples in $(K^{n\times n})^m$: the orbit degeneration preorder ($A \leq_{\mathrm{deg}} B$) defined via containment of orbit closures, and the hom-order ($3\times 3$0) determined by rank inequalities induced by substitutions from the free associative algebra $3\times 3$1. The hom-order refines orbit closure comparisons by leveraging module-theoretic invariants. Lemma 1 establishes that orbit degeneration implies hom-order, but the converse fails in general for $3\times 3$2.

Main Results

The central theorem asserts: For tuples $3\times 3$3 of $3\times 3$4 nilpotent matrices, orbit degeneration and hom-order coincide. Explicitly, $3\times 3$5 if and only if $3\times 3$6 for $3\times 3$7, where $3\times 3$8 denotes the nullcone of nilpotent matrix tuples. This establishes that degeneration order is completely characterized by rank conditions across the nullcone of $3\times 3$9 nilpotent tuples.

This result is shown to be specific to $3\times 3$0; for $3\times 3$1, counterexamples demonstrate that these two preorders do not coincide.

Module-Theoretic Framework

The paper develops a module-theoretic equivalence for the hom-order: For $3\times 3$2 in $3\times 3$3, several conditions are equivalent, including $3\times 3$4 and inequalities of dimensions of $3\times 3$5 spaces for finite-dimensional $3\times 3$6-modules. This characterization aligns orbit closure relations with properties of associated modules, linking geometric invariant theory with representation theory.

It is further demonstrated that hom-order implies a strict decrease in orbit dimension unless the orbits coincide, refining degeneration analysis beyond mere closure containment.

Explicit Orbit Classification and Degeneration

The authors provide explicit classification of $3\times 3$7-orbits of nilpotent $3\times 3$8 matrix pairs. Representatives for each orbit family are identified according to the structure of the associative subalgebra generated by the pairs, which yields six isotropy types.

The degeneration order among these orbits is mapped out, and the construction of explicit morphisms (families of group elements) is used to demonstrate orbit degenerations. The Hasse diagram captures minimal degenerations between these orbit types. Rank inequalities distinguish non-degenerate orbit pairs, confirming the tight correspondence between orbit closure and hom-order.

For $3\times 3$9, the paper verifies that degeneration and hom-order coincide via direct analysis; for general $GL_3(K)$0, arguments based on the number of generators needed for a nilpotent subalgebra and compatibility of degeneration with module-theoretic invariants extend this result.

Stratification and Additional Group Actions

The paper reviews the Hesselink stratification of the nullcone for $GL_3(K)$1 matrix pairs, partitioning the nullcone into finitely many locally closed, smooth $GL_3(K)$2-subvarieties. The degeneration order among strata is determined, and explicit orbit representatives are associated to each stratum.

The investigation is extended to group actions involving $GL_3(K)$3 and certain subgroups, leading to finitely many orbit families and explicit degeneration orders. The analysis leverages invariants such as the span dimension and the rank of matrix components to separate orbit classes and track degenerations.

Implications and Further Directions

The equivalence of orbit degeneration and hom-order for $GL_3(K)$4 nilpotent tuple orbits provides a complete and computable characterization, reducing orbit closure questions to rank inequalities. This result has practical implications for geometric complexity theory, invariant theory, and module classification, especially for small matrix sizes. Theoretical implications include the identification of a minimal dimension where degeneration order coincides with hom-order among nilpotent tuples.

The techniques could be extended to related problems in representation theory, such as orbit closure and degenerations in varieties of quiver representations and module varieties. Further study could focus on the interplay between stratifications (e.g., Hesselink, Kraft) and orbit closures for different group actions, and on explicit algorithms for testing orbit closure containment using rank invariants.

Conclusion

The paper provides a definitive analysis of degeneration order for $GL_3(K)$5 nilpotent matrix tuples, establishing that the closure relation among $GL_3(K)$6-orbits is characterized by explicit rank conditions. This bridges geometric, module-theoretic, and combinatorial perspectives, and provides both foundational and practical results for orbit classification and degeneration theory in algebraic transformation groups.