The classification of serial posets with the non-negative quadratic Tits form being principal

Vitalij M. Bondarenko ; Marina V. Styopochkina

Algebra and discrete mathematics, Tome 27 (2019) no. 2, pp. 202-211.

Résumé

Using (introduced by the first author) the method of (min, max)-equivalence, we classify all serial principal posets, i.e. the posets $S$ satisfying the following conditions: (1) the quadratic Tits form $q_S(z)\colon\mathbb{Z}^{|S|+1}\to\mathbb{Z}$ of $S$ is non-negative; (2) $\operatorname{Ker}q_S(z):=\{t\mid q_S(t)=0\}$ is an infinite cyclic group (equivalently, the corank of the symmetric matrix of $q_S(z)$ is equal to $1$); (3) for any $m\in\mathbb{N}$, there is a poset $S(m)\supset S$ such that $S(m)$ satisfies (1), (2) and $|S(m)\setminus S|=m$.

Détail

Keywords: quiver, serial poset, principal poset, minimax equivalence, one-side and two-side sums, minimax sum.
Mots-clés : quadratic Tits form, semichain

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