On algorithmic Coxeter spectral analysis of positive posets

Gąsiorek, Marcin

doi:10.1016/j.amc.2020.125507

Cited by 3 publications

(8 citation statements)

References 45 publications

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“…The main result of the paper is the following theorem that shows the correspondence between combinatorial and algebraic properties of non-negative posets of Dynkin type A n . In particular, we confirm Conjecture 6.4 stated in [8], that is, we show that in the case of non-negative posets of Dynkin type A n , there is a one-to-one correspondence between positive posets and connected digraphs whose underlying graph is a path. We give a similar description of corank 1 (i.e., principal ) posets: there is a one-to-one correspondence between such posets and connected cycles with at least two sinks.…”

Section: Introductionsupporting

confidence: 85%

“…Our description of positive posets has a direct application in the problem of Dynkin type recognition, see [8,Proposition 6.5]. This improves upon the previous best O(n 2 ) result of Makuracki-Mróz [11] and Zając [20].…”

Section: Introductionsupporting

confidence: 52%

“…Proof. Here we follow arguments given in the proof of [8,Proposition 6.7]. To calculate the number of nonisomorphic orientations of edges of P n , we consider two cases.…”

Section: Enumeration Of a N Dynkin Type Non-negative Posetsmentioning

confidence: 99%

“…The formula (5.2) describes the number of various combinatorial objects. For example: the number of linear oriented trees with n arcs or unique symmetrical triangle quilt patterns along the diagonal of an n × n square, see [12, OEIS sequence A051437].By Theorem 1.3(a), the Hasse digraph H(I) of every positive connected poset I of Dynkin type Dyn I = A n is an oriented path graph, as suggested in[8, Conjecture 6.4]. Hence, Fact 5.1 gives an exact formula for the number of all, up to the poset isomorphism, such connected posets I (see also[8, Proposition 6.7]).…”

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confidence: 99%

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Structure of non-negative posets of Dynkin type $\mathbb{A}_n$

Gąsiorek¹

2022

Preprint

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We study, in terms of directed graphs, partially ordered sets (posets) I = ({1, . . . , n}, I ) that are non-negative in the sense that their symmetric Gram matrix GI := 1 2 (CI + C tr I ) ∈ M |I| (Q) is positive semi-definite, where CI ∈ Mn(Z) is the incidence matrix of I encoding the relation I . We give a complete, up to isomorphism, structural description of connected posets I of Dynkin type Dyn I = An in terms of their Hasse digraphs H(I) that uniquely determine I. One of the main results of the paper is the proof that the matrix GI is of rank n or n − 1, i.e., every non-negative poset I with Dyn I = An is either positive or principal.Moreover, we depict explicit shapes of Hasse digraphs H(I) of all non-negative posets I with Dyn I = An. We show that H(I) is isomorphic to an oriented path or cycle with at least two sinks. By giving explicit formulae for the number of all possible orientations of the path and cycle graphs, up to the isomorphism of unlabeled digraphs, we devise formulae for the number of non-negative posets of Dynkin type An.

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Section: Introductionsupporting

confidence: 85%

Section: Introductionsupporting

confidence: 52%

“…Proof. Here we follow arguments given in the proof of [8,Proposition 6.7]. To calculate the number of nonisomorphic orientations of edges of P n , we consider two cases.…”

Section: Enumeration Of a N Dynkin Type Non-negative Posetsmentioning

confidence: 99%

mentioning

confidence: 99%

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Structure of non-negative posets of Dynkin type $\mathbb{A}_n$

Gąsiorek¹

2022

Preprint

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“…We note that the Coxeter spectral classification of finite posets, up to the Gram Z-congruences defined in Section 2, grew up from many different branches of mathematics and computer science and is successfully applied in Lie theory, Diophantine geometry, algebraic combinatorics, representation theory, matrix analysis, graph theory, combinatorial and graph algorithms, singularity theory, and related areas. The reader is referred to [5,6,12] and [13,Section 6.1.2] for a more detailed discussion.…”

Section: Introductionmentioning

confidence: 99%

A Coxeter type classification of Dynkin type $\mathbb{A}_n$ non-negative posets

Gąsiorek¹

2022

Preprint

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We continue the Coxeter spectral analysis of finite connected posets I that are nonnegative in the sense that their symmetric Gram matrix G I := 1 2 (C I + C tr I ) ∈ M n (Q) is positive semi-definite of rank n − crk I ≥ 0, where C I ∈ M n (Z) is the incidence matrix of I encoding the relation I . We extend the results of [Fundam. Inform., 139.4(2015), 347-367] and give a complete Coxeter spectral classification of finite connected posets I of Dynkin type A m .We show that such posets I, with |I| > 1, yield exactly ⌊ n 2 ⌋ Coxeter types, one of which describes the positive (i.e., crk I = 0) ones. We give an exact description and calculate the number of posets of every type. Moreover, we prove that, given a pair of such posets I and J, the incidence matrices C I and C J are Z-congruent if and only if specc I = specc J , and present deterministic algorithms that calculate a Z-invertible matrix defining such a Z-congruence in a polynomial time.

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Applications of mesh algorithms and self-dual mesh geometries of root Coxeter orbits to a Horn-Sergeichuk type problem

Simson

Zając

2022

Linear Algebra and its Applications

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On algorithmic Coxeter spectral analysis of positive posets

Cited by 3 publications

References 45 publications

Structure of non-negative posets of Dynkin type $\mathbb{A}_n$

Structure of non-negative posets of Dynkin type $\mathbb{A}_n$

A Coxeter type classification of Dynkin type $\mathbb{A}_n$ non-negative posets

Applications of mesh algorithms and self-dual mesh geometries of root Coxeter orbits to a Horn-Sergeichuk type problem

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