Description of posets critical with respect to the nonnegativity of the quadratic Tits form

Bondarenko, V. S.; Степочкина, М. В.

doi:10.1007/s11253-009-0245-6

Cited by 19 publications

(4 citation statements)

References 6 publications

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“…is weakly positive definite by (a), but it is not nonnegative definite since q(−1, 2, 2, −2, −2, −2) = −1. Bondarenko and Stepochkina [3] (see also [4]) gave a list of posets with positive definite Tits form; it consists of four infinite series and 108 posets defined up to duality; this list was constructed in an alternative way in [9]. In contrast to this, we prove in the next theorem that "positive definite" and "nonnegative definite" can be replaced in Theorem 1 by "weakly positive definite" and "weakly nonnegative definite".…”

Section: Systems Of Subspaces Of a Vector Spacementioning

confidence: 87%

Systems of subspaces of a unitary space

Bondarenko

Futorny

Klimchuk

et al. 2013

Linear Algebra and its Applications

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Section: Systems Of Subspaces Of a Vector Spacementioning

confidence: 87%

Systems of subspaces of a unitary space

Bondarenko

Futorny

Klimchuk

et al. 2013

Linear Algebra and its Applications

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“…( ) = 0} (14) is an infinite cyclic subgroup of Z . Following the main idea of the Coxeter spectral analysis of acyclic edge-bipartite graphs (signed graphs) presented in [3,4], we study finite posets (with a fixed numbering = { 1 , .…”

Section: Definition 2 (A)mentioning

confidence: 99%

A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots

Simson

Zając

2013

International Journal of Mathematics and Mathematical Sciences

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Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posetsJ≡(J,⪯). One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219–225]. We are mainly interested in a Coxeter spectral classification of posetsJsuch that the symmetric Gram matrixGJ:=(1/2)[CJ+CJtr]∈𝕄J(ℚ)is positive semidefinite, whereCJ∈𝕄J(ℤ)is the incidence matrix ofJ. Following the idea of Drozd mentioned earlier, we associate toJits Coxeter matrixCoxJ:=-CJ·CJ-tr, its Coxeter spectrumspeccJ, a Coxeter polynomialcoxJ(t)∈ℤ[t], and a Coxeter number cJ. In caseGJis positive semi-definite, we also associate toJa reduced Coxeter number čJ, and the defect homomorphism∂J:ℤJ→ℤ. In this case, the Coxeter spectrumspeccJis a subset of the unit circle and consists of roots of unity. In caseGJis positive semi-definite of corank one, we relate the Coxeter spectral properties of the posetsJwith the Coxeter spectral properties of a simply laced Euclidean diagramDJ∈{𝔻̃n,𝔼̃6,𝔼̃7,𝔼̃8}associated withJ. Our aim of the Coxeter spectral analysis of such posetsJis to answer the question when the Coxeter typeCtypeJ:=(speccJ,cJ, čJ)ofJdetermines its incidence matrixCJ(and, hence, the posetJ) uniquely, up to aℤ-congruency. In connection with this question, we also discuss the problem studied by Horn and Sergeichuk [Linear Algebra Appl. 389(2004), 347–353], if for anyℤ-invertible matrixA∈𝕄n(ℤ), there isB∈𝕄n(ℤ)such thatAtr=Btr·A·BandB2=Eis the identity matrix.

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“…Among others, some of the recent examples of such an algorithmic approach to mathematical problems are the following: a solution of a class Birkhoff type problems [26], the mesh algorithms for describing mesh root systems for a class of non-negative Diophantine equations [22], algorithmic Coxeter spectral study of edge-bipartite graphs and posets and quadratic forms [4,5,10,11,14,15,23,25,27], and the decomposition and classification problems in module (representation) theory and matrix problems [2,8,9,12,13,19,21], see also [17,Section 5] and [24,Remark 4.2].…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Algorithms for Computing Degenerations of Modules of Finite Dimension

Mróz

Zwara

2014

Fundamenta Informaticae

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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 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.

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Description of posets critical with respect to the nonnegativity of the quadratic Tits form

Cited by 19 publications

References 6 publications

Systems of subspaces of a unitary space

Systems of subspaces of a unitary space

A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots

Combinatorial Algorithms for Computing Degenerations of Modules of Finite Dimension

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