Antimonotone quadratic forms and partially ordered sets

Abstract:

Representations of partially ordered sets (posets) and quivers are an important part of the theory of matrix problems and algebra representations. Along with chains (linearly ordered sets), a special role is played by certain special posets; in this paper it is shown that they are in one-to-one correspondence with the rational numbers that are greater than or equal to $1$.

A wattle $\langle n_1 ,\dots ,n_t\rangle$ is a union of nonintersecting chains $Z_i$ $(|Z_i|=n_i)$ such that the minimal element of $Z_i$ is smaller than the maximal element of $Z_{i+1}$ $(i=1 ,\dots ,t-1)$ (and these are the only possible comparisons). The known lists of critical (i.e., minimal) infinitely representable and wild posets consist of cardinal chains, with the exception of one poset in the first list (namely, $\langle 2,2\rangle +Z_4$) and one in the second (namely, $\langle 2,2\rangle +Z_5)$. At the same time, the authors have assigned a rational number $P(S)$ to each poset $S$ in such a way that $P(S)<4$ if and only if $S$ is finitely representable and $P(S)=4$ if and only if $S$ is tame. A poset $S$ is said to be $P$-faithful if $P(S’)<P(S)$ whenever $S’\subset S$.

From the work of Zel′dich, Sapelkin, and the authors it follows that the $P$-faithful posets are cardinal sums of $r$-sets, i.e., they are wattles of a special type (chains can be regarded as a partial case of $r$-sets).

In the present paper, the notion of an antimonotone poset is introduced, and a criterion for a poset to be antimonotone is presented under the assumption that the quadratic form $\sum _{s_i\le s_j}x_ix_j$ $(S=\{s_1 ,\dots ,s_n\})$ is positive semidefinite. At the same time, we manage to substantially simplify the proof of the criterion for a poset to be $P$-faithful, avoiding an item-by-item examination of several dozens of various cases. Also, simple explicit formulas for calculation of $P(S)$ are obtained, which lead in an elementary way to the lists of critical posets (originally, they arose as a result of a cumbersome and complex argument).