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• W2024955032 abstract "Following an idea of spectral graph theory, we give several characterizations of finite one-peak posets I with almost P-critical quadratic Tits form qˆI:ZI→Z by applying linear algebra methods, combinatorial algorithms, an old idea of S.A. Ovsienko and recent results by V.M. Bondarenko and M.V. Stepochkina. Our study is mainly motivated by an important application of the quadratic Tits form of a poset in constructing linear algebra invariants that measure a geometric complexity of Nazarova–Roiter matrix problems over a field K. In particular, our study is inspired by a well-known result of Ju.A. Drozd explained in Introduction. One of our main results asserts that, given a finite poset I={1,…,n,⁎}=T∪{⁎} such that ⁎ is its unique maximal element and qˆI is almost P-critical, we have: the form qˆI is non-negative and the subgroup KerqˆI:={v∈Zn+1;qˆI(v)=0} of ZI≡Zn+1 is generated by a vector h=(h1,…,hn,h⁎), with h1≠0,…,hn≠0, 5⩽|I|⩽9 and the Coxeter polynomial coxI(t)∈Z[t] coincides with the Coxeter polynomial FΔI(t) of a uniquely determined Euclidean diagram ΔI∈{D˜4,E˜6,E˜7,E˜8}, the Z-bilinear Tits form bˆI:Zn+1×Zn+1→Z of I is Z-equivalent to the Gram Z-bilinear form bΔI:Zn+1×Zn+1→Z of the Euclidean diagram ΔI, there is a Z-invertible matrix C such that CˆItr=Ctr⋅CˆI⋅C and C2=E, where CˆI∈Mn+1(Z) is the Tits matrix of I, and I is one of the posets L1,…,L132 listed in the paper." @default.
• W2024955032 created "2016-06-24" @default.
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• W2024955032 date "2014-03-01" @default.
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• W2024955032 title "Coxeter spectral classification of almost TP -critical one-peak posets using symbolic and numeric computations" @default.
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• W2024955032 doi "https://doi.org/10.1016/j.laa.2013.12.018" @default.
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