Andrzej Mróz

Nicolaus Copernicus University | umk · Faculty of Mathematics and Computer Science

We present a general solution of the isomorphism and multiplicity problems, restricted to the class of all modules lying in homogeneous tubes, for tame algebras (Theorem 2.4). We introduce a notion of the {\em characteristic polynomial} of a module, which plays an analogous role as in the classical situation. This notion uses a Smith form $\Delta(\...

We introduce and study in detail so-called circulant (Coxeter-periodic) elements and circulant families in a bilinear lattice as well as their dual versions, called anti-circulant. We show that they form a natural environment for a systematic explanation of certain cyclotomic factors of the Coxeter polynomial of and in consequence, of Coxeter polyn...

Cartan matrices and quasi-Cartan matrices play an important role in such areas like Lie theory, representation theory and algebraic graph theory. It is known that each (connected) positive definite quasi-Cartan matrix A ∈ Mn(Z) is Z-equivalent with the Cartan matrix of a Dynkin diagram, called the Dynkin type of A. We present a symbolic, graph-theo...

We study integral quadratic forms in the sense of Roiter, that is, quadratic forms whose integer coefficients satisfy certain divisibility condition assuring that the associated Weyl group is integral. Such forms are known to be useful for characterizing classes of finite-dimensional algebras and Lie algebras. We present a solution of the problem c...

We formulate a simple model for the bounded derived category of gentle algebras in terms of marked ribbon graphs and their walks, in order to analyze indecomposable objects, Auslander-Reiten triangles and homological bilinear forms, and to provide some relevant derived invariants in a graph theoretic setting. Among others, we exhibit the non-negati...

Bidirected graphs are directed multigraphs admitting arrows having two starting or two ending points. With arbitrary such graph we associate a non-negative integral quadratic form as a so-called incidence form. We show that each non-negative form of Dynkin type A, D or C arises in this way, and that all certain roots of an integral quadratic form c...

Cartan matrices, quasi-Cartan matrices and associated upper triangular Gram matrices control important combinatorial aspects of Lie theory and representation theory of associative algebras. We provide a graph theoretic proof of the fact that the absolute values of the coefficients of a non-negative quasi-Cartan matrix A as well as of its (minimal)...

For standard algorithms verifying positive definiteness of a matrix $A\in\mathbb{M}_n(\mathbb{R})$ based on Sylvester's criterion, the computationally pessimistic case is this when $A$ is positive definite. We present two algorithms realizing the same task for $A\in\mathbb{M}_n(\mathbb{Z})$, for which the case when $A$ is positive definite is the o...

Cartan matrices, quasi-Cartan matrices and associated inte-gral quadratic forms and root systems play an important role in such areas like Lie theory, representation theory and alge-braic graph theory. We study quasi-Cartan matrices by means of the inflation algorithm, an idea used in Ovsienko’s classical proof of the classification of positive def...

We study edge-bipartite graphs (bigraphs), a class of signed graphs, by means of the inflation algorithm which relies on performing certain elementary transformations on a given bigraph ∆, or equivalently, on the associated integral quadratic form q ∆ : Z n → Z, preserving Gram Z-congruence. The ideas are inspired by classical results of Ovsienko a...

In this two parts article with the same main title we study a problem of Coxeter-Gram spectral analysis of edge-bipartite graphs (bigraphs), a class of signed graphs. We ask for a criterion deciding if a given bigraph ∆ is weakly or strongly Gram-congruent with a graph. The problem is inspired by recent works of Simson et al. started in [SIAM J. Di...

We study the concept of the Coxeter energy of graphs and digraphs (quivers) as an analogue of Gutman's adjacency energy, which has applications in theoretical chemistry and is a recently widely investigated graph invariant. Coxeter energy of a (di)graph G is defined to be the sum of the absolute values of all complex eigenvalues of the Coxeter matr...

Let Λ be a k-algebra of finite global dimension. We study tubular families in the Auslander–Reiten quiver of the bounded derived category Db(Λ) satisfying certain natural axioms. In particular, we precisely describe their influence on the cyclotomic factors of the Coxeter polynomial χΛ of Λ and discuss several numerical limitations for their possib...

We present combinatorial algorithms for solving three problems that appear in the study of the degeneration order ≤ for the variety of finite-dimensional modules over a k-algebra A, where M ≤ N means that a module N belongs to an orbit closure \overline{O(M)} of a module M in the variety of A-modules. In particular, we introduce algorithmic techniq...

We give an algorithmic description of matrix bimodules parametrizing all indecomposable homogeneous Λ-modules with a fixed slope q over a tubular canonical algebra Λ, for all possible slopes q (Main Theorem 3.3). A crucial role in this description is played by universal extensions of bimodules and their nice properties (Theorems 3.1 and 3.2).

We give a description of matrix bimodules parametrizing all indecomposable homogeneous Λ-modules with a fixed integral slope over a tubular canonical algebra Λ, for all possible integers (Theorem 4.1). An important role in the first step of this description (Theorem 2.4) is played by the translation of the shift functor for coherent sheaves over th...

We study the complexity of Bongartz's algorithm for determining a maximal common direct summand of a pair of modules $M,N$ over $k$-algebra $\La$; in particular, we estimate its pessimistic computational complexity $\CO(rm^6n^2(n+m\log n))$, where $m=\dimk M\leq n=\dimk N$ and $r$ is a number of common indecomposable direct summands of $M$ and $N$....

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...

This is the addendum to the paper "On the Multiplicity Problem and the Isomorphism Problem for the Four Subspace Algebra" Communications in Algebra, 40:6 (2012), 2005-2036 (DOI: 10.1080/00927872.2011.570830). We give here the full proof of Proposition 3.3, describing the formulas for the dimensions of the homomorphism spaces to indecomposable modul...

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 matricesM with coefficients in a field K. One of the main aims is to reduce M to its special canonical form M1 with respect to...

Let � be the four subspace algebra. We show that for any �-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 �-...

Let Λ be a tubular canonical algebra of quiver type. We describe an algorithm, which for numerical data computes all regular exceptional Λ-modules, or more generally all indecomposable modules in exceptional tubes. The input for the algorithm is a quadruple consisting of the slope, the number of the tube, the quasi-socle and the quasi-length, the o...

Given a pair M;M0 of finite-dimensional modules over a string special biserial algebra �, a fully verifiable criterion, expressed in terms of a finite set of simple linear algebra invariants, deciding if M and M0 lie in the same orbit in module variety, equivalently, if M and M0 are isomorphic, is formulated and proved.

Given a module M over a domestic canonical algebra A and a classifying set X for the indecomposable A-modules, the problem of determining the vector m(M)=(mx)x∈X ∈ ℕX such that (Formula presented) is studied. A precise formula for dimk HomΛ (M,X), for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between...

Given a pair M,M ' of finite-dimensional modules over a domestic canonical algebra Λ, we give a fully verifiable criterion, in terms of a finite set of simple linear algebra invariants, deciding if M and M ' lie in the same orbit in the module variety, or equivalently, if M and M ' are isomorphic.

Given a module M over an algebra Λ and a complete set X of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector m(M) = (mX)X∈X ∈ ℕX such that (Formula Presented) is studied. A general method of finding the vectors m(M) is presented (Corollary 2.1, Theorem 2.2 and Corollary 2.3). It is discussed and applied in pract...