Combinatorial Algorithms for Computing Degenerations of Modules of Finite Dimension

Article type: Research Article

Authors: Mróz, Andrzej | Zwara, Grzegorz

Affiliations: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland. amroz@mat.umk.pl; gzwara@mat.umk.pl

Note: [] Address for correspondence: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland.

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.

DOI: 10.3233/FI-2014-1057

Journal: Fundamenta Informaticae, vol. 132, no. 4, pp. 519-532, 2014