Andrzej Mróz

On the Computational Complexity of Bongartz's Algorithm

We study the complexity of Bongartzs algorithm for determining a maximal common direct summand of a pair of modules M, N over k-algebra Λ; in particular, we estimate its pessimistic computational complexity Orm6n2n + m log n, where m = dimkM ≤ n = dimkN and r is a number of common indecomposable direct summands of M and N. We improve the algorithm to another one of complexity Orm4n2n+m log m and we show that it applies to the isomorphism problem having at least an exponential complexity in a direct approach. Moreover, we discuss a performance of both algorithms in practice and show that the “average” complexity is much lower, especially for the improved one which becomes a part of QPA package for GAP computer algebra system.

Tree Matrices and a Matrix Reduction Algorithm of Belitskii

Inspired by the bimodule matrix problem technique and various classification problems in poset representation theory, finite groups and algebras, we study the action of Belitskii algorithm on a class of square n by n block matrices M with coefficients in a field K. One of the main aims is to reduce M to its special canonical form M∞ with respect to the conjugation by elementary transformations defined by a class of matrices chosen in a subalgebra of the full matrix algebra

Experiences in Symbolic Computations for Matrix Problems

\mathbb{M}_n

On the Multiplicity Problem and the Isomorphism Problem for the Four Subspace Algebra

(K). The algorithm can be successfully applied in the study of indecomposable linear representations of finite posets by a computer search using numeric and symbolic computation. We mainly study the case when the di-graph (quiver) associated to the output matrix M∞ of the algorithm is a disjoint union of trees. We show that exceptional representations of any finite poset are determined by tree matrices. This generalizes a theorem of C.M. Ringel proved for linear representations of di-graphs.

Combinatorial Algorithms for Computing Degenerations of Modules of Finite Dimension

We review our recent results concerning several computer algebra aspects of determining canonical forms, performing a decomposition and deciding the isomorphism question for matrix problems. We consider them in the language of finite dimensional modules over algebra and the language of square block matrices with an action of elements from some sub algebra of the full matrix algebra. We present an efficient (polynomial-time) improvement of classical Bongartzs algorithm for determining a maximal common direct summand of modules, and its application to solving the isomorphism problem. The improved algorithm recently became a part of QPA package ver. 1.07 for GAP. We also study the behaviour of Belitskiis algorithm for determining certain canonical form on a class of square block matrices, especially for matrix problems associated with a poset. Both problems can be considered as a highly generalized classical Jordan problem for square matrices.

Effective Nondeterministic Positive Definiteness Test for Unidiagonal Integral Matrices

Let Λ be the four subspace algebra. We show that for any Λ-module M there exists an algorithm (up to the problem of finding roots of the so-called characteristic polynomial of M) with relatively low polynomial complexity of determining multiplicities of all direct summands of M. Moreover, we give a fully algorithmic criterion for deciding if two Λ-modules M and N are isomorphic.

Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition II. Coxeter Type Study*

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

Tubes in derived categories and cyclotomic factors of the Coxeter polynomial of an algebra

\overline{\cal{O}(M)}

On a separation of orbits in the module variety for string special biserial algebras

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.

The multiplicity problem for indecomposable decompositions of modules over a finite-dimensional algebra. Algorithms and a computer algebra approach

For standard algorithms verifying positive definiteness of a matrix A ∈ Mn(R) based on Sylvester’s criterion, the computationally pessimistic case is this when A is positive definite. We present an algorithm realizing the same task for A ∈ Mn(Z), for which the case when A is positive definite is the optimistic one. The algorithm relies on performing certain edge transformations, called inflations, on the signed graph (bigraph) Δ = Δ(A) associated with A. We provide few variants of the algorithm, including Las Vegas type randomized ones (with precisely described maximal number of steps). The algorithms work very well in practice, in many cases with a better speed than the standard tests. On the other hand, our results provide an interesting example of an application of symbolic computing methods originally developed for different purposes, with a big potential for further generalizations in matrix problems.