A classification of two-peak sincere posets of finite prinjective type and their sincere prinjective representations

Kosakowska, Justyna

doi:10.4064/cm87-1-3

Cited by 8 publications

(18 citation statements)

References 13 publications

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“…Let K be an arbitrary field. We prove Theorem 2.1 in Section 7 by developing the methods introduced in [14] and in Section 6 of the present paper.…”

Section: The Main Classification Theoremmentioning

confidence: 99%

“…Below we collect some properties of positive roots of integral quadratic forms. For the proofs the reader is referred to [10], [14] and [18].…”

Section: Tables 22mentioning

confidence: 99%

“…In the present paper we continue the study of sincere posets started in [14], where the motivation for the study of prinjective modules is presented. For the motivation see also [2], [28]- [30].…”

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confidence: 93%

“…A complete list of 60 sincere two-peak posets of finite prinjective type and their 328 sincere prinjective representations is presented in [14].…”

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confidence: 99%

“…We construct inductively sincere posets I of finite prinjective type such that |I| = n + 1 from sincere posets J of finite prinjective type such that |J| = n. Here we develop the technique introduced in [14].…”

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confidence: 99%

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Sincere posets of finite prinjective type with three maximal elements and their sincere prinjective representations

Kosakowska

2002

Colloq. Math.

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Abstract. Assume that K is an arbitrary field. Let (I, ) be a poset of finite prinjective type and let KI be the incidence K-algebra of I. A classification of all sincere posets of finite prinjective type with three maximal elements is given in Theorem 2.1. A complete list of such posets consisting of 90 diagrams is presented in Tables 2.2. Moreover, given any sincere poset I of finite prinjective type with three maximal elements, a complete set of pairwise non-isomorphic sincere indecomposable prinjective modules over KI is presented in Tables 8.1. The list consists of 723 modules.0. Introduction. Throughout this paper I = (I, ) is a finite poset (i.e. partially ordered set) with partial order . We write i ≺ j if i j and i = j. For simplicity we write I instead of (I, ). The poset I is said to be connected if I is not the union of two proper subposets I 1 , I 2 such that I 1 ∩ I 2 = ∅. Throughout the paper all posets are assumed to be connected. Following [21] we denote by max I the set of all maximal elements of I (called peaks of I). The poset I is called an r-peak poset if |max I| = r, where |X| denotes the cardinality of the set X. A subposet J of I is said to be a peak subposet if J ∩ max I = max J.Throughout the paper, K is a field. We denote by KI the incidence K-algebra of the poset I, that is, KI is the K-subalgebra of the full I × I matrix algebra M I (K) consisting of all matrices [λ ij ] in M I (K) such that λ ij = 0 if i j in (I, ) (see [20], [21]). Given j ∈ I we denote by e j ∈ KI the standard primitive idempotent corresponding to j.A right KI-module X is identified with a systemwhere X i = Xe i is a finite-dimensional vector space over K and j h i : X i → X j , i ≺ j, are K-linear maps such that i h i = id and t h j · j h i = t h i for all i ≺ j ≺ t in I. Such systems are called K-linear representations of

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“…Let K be an arbitrary field. We prove Theorem 2.1 in Section 7 by developing the methods introduced in [14] and in Section 6 of the present paper.…”

Section: The Main Classification Theoremmentioning

confidence: 99%

“…Below we collect some properties of positive roots of integral quadratic forms. For the proofs the reader is referred to [10], [14] and [18].…”

Section: Tables 22mentioning

confidence: 99%

mentioning

confidence: 93%

“…A complete list of 60 sincere two-peak posets of finite prinjective type and their 328 sincere prinjective representations is presented in [14].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Sincere posets of finite prinjective type with three maximal elements and their sincere prinjective representations

Kosakowska

2002

Colloq. Math.

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Coxeter spectral classification of almost TP -critical one-peak posets using symbolic and numeric computations

Polak

Simson

2014

Linear Algebra and its Applications

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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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A classification of two-peak sincere posets of finite prinjective type and their sincere prinjective representations

Cited by 8 publications

References 13 publications

Sincere posets of finite prinjective type with three maximal elements and their sincere prinjective representations

Sincere posets of finite prinjective type with three maximal elements and their sincere prinjective representations

Coxeter spectral classification of almost TP -critical one-peak posets using symbolic and numeric computations

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

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