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

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 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 such that ⁎ is its unique maximal element and is almost P-critical, we have: [(i)] the form is non-negative and the subgroup of is generated by a vector , with , [(ii)] and the Coxeter polynomial coincides with the Coxeter polynomial of a uniquely determined Euclidean diagram , [(iii)] the -bilinear Tits form of I is -equivalent to the Gram -bilinear form of the Euclidean diagram , [(iv)] there is a -invertible matrix C such that and , where is the Tits matrix of I, and [(v)] I is one of the posets listed in the paper. In particular, we get in an alternative way the list of P-critical posets T (in the form ) presented by V.M. Bondarenko and M.V. Stepochkina [6], and we determine their Coxeter–Euclidean types. There are 115 one-peak posets , with P-critical, and 17 exceptional posets , with almost P-critical, but not P-critical. Our lists are obtained by using computer algebra tools; mainly symbolic computation in Maple and numeric computation in C#. [Copyright &y& Elsevier]