Combinatorial Algorithms for Computing Degenerations of Modules of Finite Dimension

Abstract

We present combinatorial algorithms for solving three problems that appear in the study of the degeneration order ≤_deg for the variety of finite-dimensional modules over a k-algebra Λ, where M ≤_deg N means that a module N belongs to an orbit closure $\overline{\cal{O}(M)}$ of a module M in the variety of Λ-modules. In particular, we introduce algorithmic techniques for deciding whether or not the relation M ≤_deg N holds and for determining all predecessors (resp. succesors) of a given module M with respect to ≤_deg. The order ≤_deg plays an important role in modern algebraic geometry and module theory. Applications of our technique and experimental tests for particular classes of algebras are presented. The results show that a computer algebra technique and algorithmic computer calculations provide important tools in solving theoretical mathematics problems of high computational complexity. The algorithms are implemented and published as a part of an open source GAP package called QPA.