P-critical integral quadratic forms and positive unit forms: An algorithmic approach

2015

Colloq. Math.

The incidence coalgebras K I of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form q • : Z (I) → Z, where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category K I-comod of finite-dimensional left K I-modules is equivalent to the tameness of the category K I-Comod fc of finitely copresented left K I-modules. Hence, the tame-wild dichotomy for the coalgebras K I is deduced. Moreover, we prove that for an interval finite A * m -free poset I the incidence coalgebra K I is of tame comodule type if and only if the quadratic form q • is weakly non-negative. Finally, we give a complete list of all infinite connected interval finite A * m -free posets I such that K I is of tame comodule type. In this case we prove that, for any pair of finite-dimensional left K I-comodules M and N , bK I (dim M, dim N ) = ∞ j=0 (−1) j dimK Ext j K I (M, N ), where bK I : Z (I) × Z (I) → Z is the Euler Z-bilinear form of I and dim M , dim N are the dimension vectors of M and N .

Section: Preliminaries On Incidence Coalgebras and Their Comodulesmentioning

confidence: 85%

Incidence coalgebras of interval finite posets of tame comodule type

Leszczyński

2015

Colloq. Math.

“…andΦ (ĥ ) =ĥ . This means that̂is principal, but notcritical; see [44]. One easily shows that the reduced Coxeter number of equalsč = 2 and the Tits defect̂: Z 5 → Z of is given bŷ( ) = −( 1 + 2 + 3 + 4 ), becauseΦ ̸ = and Φ 2 ( ) = +̂( ) ⋅ĥ , for any ∈ Z 5 .…”

Section: An Examplementioning

confidence: 98%

“…It follows that is nonnegative and Ker = Z ⋅ h, where h = (1, 1, 1, 1, 1, 1, 1); is critical in the sense of Ovsienko [24]; see also [38,44]. Note that the Euler matrix = −1 of and the inverse of the Coxeter-Euler matrix Cox := − −1 ⋅ tr have the forms…”

Section: Principal Posetsmentioning

confidence: 99%

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

International Journal of Mathematics and Mathematical Sciences

Zając

2013

Self Cite

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.

“…Our main aim in this section is to show that the computation of the set CGpol + n of all polynomials cox Δ (t), with positive edge-bipartite graphs Δ in UBigr n , n ≥ 2, and the proof of Theorem 2.3 reduces to the computation of Gl(n, Z) D -orbits on the set Mor D of matrix morsifications for simply laced Dynkin diagrams D. Using this idea, we construct algorithmic procedures for the Coxeter spectral analysis of loop-free edge-bipartite graphs in UBigr n , by applying symbolic and numerical computations in Linux, Maple and C++, with GNU Scientific Library. Here we mainly apply the technique and results given in [8]- [10], [15]- [16], [19]- [22], and [25]- [27].…”

Section: A Reduction To Matrix Morsificationsmentioning

confidence: 99%

On Coxeter Type Classification of Loop-Free Edge-Bipartite Graphs and Matrix Morsifications

Bocian

Felisiak

2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

2013

Self Cite

We continue and complete a Coxeter spectral study (presented in our talk given in SYNASC11 and SYNASC12) of the root systems in the sense of Bourbaki, the mesh geometries Γ(RΔ, ΦA) of roots of Δ in the sense of [J. Pure Appl. Algebra, 215 (2010), 13-34], and matrix morsifications A ∈ MorΔ, for simply laced Dynkin diagrams Δ ∈ {An, Dn, E6, E7, E8}. Here we report on algorithmic and morsification technique for the Coxeter spectral analysis of connected loop-free edge-bipartite graphs Δ with n ≥ 2 vertices by means of the Coxeter matrix CoxΔ ∈ Mn(Z), the Coxeter spectrum specc Δ , and an inflation algorithm associating to any connected loop-free positive bigraph Δ a simply laced Dynkin diagram DΔ, and defining a Z-congruence of the symmetric Gram matrices GΔ and GDΔ. We also present a computer aided technique that allows us to construct a Z-congruence of the non-symmetric Gram matricesǦΔ andǦ Δ , if the Coxeter spectra specc Δ and specc Δ coincide. A complete Coxeter spectral classification of positive edge-bipartite graphs Δ of Coxeter-Dynkin types DΔ ∈ {An, Dn, E6, E7}, with n ≤ 7, is obtained by a reduction to computer calculation of Gl(n, Z)DΔ-orbits in the set MorDΔ, where Gl(n, Z)DΔ is the isotropy group of the Dynkin diagram DΔ.