Stanisław Kasjan

Tame prinjective type and Tits form of two-peak posets I☆

Abstract The main aim of this paper is to give a simple criterion for a finite poset I with two maximal elements to have the category I -spr of socle projective representations of tame representation type. Our main result is Theorem 1 which asserts that for any upper chain reducible poset I with two maximal elements (see Definition 8) the category I -spr is of tame representation type if and only if the Tits quadratic form q I : Q I → Q (1.1) of I is weakly non-negative, or equivalently, if and only if I does not contain as a peak subposet any of the one-peak posets N 1 ∗ ,…, N 6 ∗ of Nazarova presented in Theorem 1 or any of the 41 two-peak posets listed in Table 1.

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

Galois covering technique and tame non-simply connected posets of polynomial growth

\mathbb{M}_n

Experiences in Symbolic Computations for Matrix Problems

(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.

On Generalized CB-Degenerations of Algebras

Abstract A combinatorial criterion for polynomial growth of partially ordered sets which are not simply connected is given. It is obtained by use of Galois covering techniques applied to poset representations. For this purpose a relative version of the basic Galois covering theory of k -categories is developed in the paper. The remaining part of the proof of the main result is based on standard methods like “peak reductions” and Splitting Lemma used in the theory of multipeak posets.

A Note on Periodicity in Representation Theory of Finite Dimensional Algebras

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.

Tameness criterion for posets with zero-relations and three-partite subamalgams of tiled orders

We introduce a concept generalizing classical degenerations of algebras (defined by structure constants) and Crawley-Boevey degenerations introduced in [3]. We prove that if A 0 is such a generalized degeneration of A 1 and the algebras have equal dimensions, then A 0 is a degeneration of A 1 in the classical sense.

A criterion for polynomial growth of a-free two-peak posets of tame prinjective type *

Let Λ be an algebra over an algebraically closed field K of infinite transcendence degree over its prime subfield. We prove in particular that if every s-dimensional indecomposable nonprojective Λ-module is Ω-periodic (resp. DTr-periodic) then there exists a common bound for the Ω-periods (resp. DTr-periods) of s-dimensional indecomposable nonprojective Λ-modules.

0-1 sequences of the Thue-Morse type and Sarnak’s conjecture

A criterion for tame prinjective type for a class of posets with zerorelations is given in terms of the associated prinjective Tits quadratic form and a list of hypercritical posets. A consequence of this result is that if Λ• is a three-partite subamalgam of a tiled order then it is of tame lattice type if and only if the reduced Tits quadratic form qΛ• associated with Λ • in [26] is weakly non-negative. The result generalizes a criterion for tameness of such orders given by Simson [28] and gives an affirmative answer to [28, Question 4.7].

Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops, I. Mesh Root Systems

It is well known from the results of L, A. Nazarova and A. G. Zavadskij [18], [19], see also [25, Chapter 15], that a poset J with one maximal element is of tame prinjective type and of polynomial growth if and only if J does not contain neither any of the Nazarovas hypercritical posets (1,1,1,1,1)*, (1,1,1,2)*,(2,2,3)*, (1,3,4)*,(W,5)*,(1,2,6)* nor the Nazarova-Zavadskij poset (NZ)* (see Table 1 below). In the present paper we extend this result to a class of posets with two maximal elements. We show that Ã-free poset with two maximal elements is of tame representation type and of polynomial growth if and only if the Tits quadratic form qs → Z (see (1.7) below) associated with J is weakly non-negative and J does not contain any of the six posets listed in Table 1 as a peak subposet.