Justyna Kosakowska

Inflation Algorithms for Positive and Principal Edge-bipartite Graphs and Unit Quadratic Forms

We describe combinatorial algorithms that compute the Dynkin type (resp. Euclidean type) of any positive (resp. principal) unit quadratic form q :

On tits form and prinjective representations of posets of finite prinjectivetype

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Operations on arc diagrams and degenerations for invariant subspaces of linear operators

n →

Operations on arc diagrams and degenerations for invariant subspaces of linear operators. Part II

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Hereditary coalgebras and representations of species

and of any positive (resp. principal) edge-bipartite connected graph Δ. The study of the problem is inspired by applications of the algorithms in the representation theory, in solving a class of Diophantine equations, in the study of mesh geometries of roots, in the spectral analysis of graphs, and in the Coxeter-Gram classification of edge-bipartite graphs.

Lie algebras associated with quadratic forms and their applications to Ringel-Hall algebras

The main aim of this paper is to gove an extencion of the charcterisation given in [18] of finite posets J having only finitely many isomorphism classes of indecomposable socle projective K-linear representations over a given field K. The characterisation is given in Theprem 1.2 and in Theorem 4.1 in terms of the Tits quadratic form associated to J, in terms of the endomorphism rings and self-extensions of indecomposable representations, and in terms of the infinitd radical of the category prin(KJ). A converse to a theorem of Bongartz [4] is given in Corollary 3.5.

A Horizontal Mesh Algorithm for a Class of Edge-bipartite Graphs and their Matrix Morsifications

Abstract: We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are algebraic groups acting on these varieties. We give dimensions of orbits of these actions. Moreover, a combinatorial characterization of the partial order given by degenerations is described. MSC 2010: Primary: 14L30, 16G20, Secondary: 16G70, 05C85, 47A15

Arc diagram varieties

ABSTRACT For a partition β, denote by Nβ the nilpotent linear operator of Jordan type β. Given partitions β, γ, we investigate the representation space of all short exact sequences where α is any partition with each part at most 2. Due to the condition on α, the isomorphism type of a sequence ℰ is given by an arc diagram Δ; denote by 𝕍Δ the subset of of all sequences isomorphic to ℰ. Thus, the variety carries a stratification given by the subsets of type 𝕍Δ. We compute the dimension of each stratum and show that the boundary of a stratum 𝕍Δ consists exactly of those where Δ′ is obtained from Δ by a non-empty sequence of arc moves of five possible types (A)–(E). The case where all three partitions are fixed has been studied in [3, 5]. There, arc moves of types (A)–(D)suffice to describe the boundary of a 𝕍Δ in . Our fifth move (E), “explosion,” is needed to break up an arc into two poles to allow for changes in the partition α.