Tame prinjective type and Tits form of two-peak posets. I
Let \(S\) be a finite partially ordered set with two maximal elements (a two-peak poset) and let \(k\) be an algebraically closed field. The authors study the category of prinjective modules over the incidence algebra \(kS\) of the poset \(S\) over \(k\), as defined by \textit{J. A. de la Peña} and \textit{D. Simson} [Trans. Am. Math. Soc. 329, No. 2, 733-753 (1992; Zbl 0789.16010)]. The Tits rational quadratic form of \(S\) for the category of prinjective modules was defined by \textit{D. Simson} [J. Pure Appl. Algebra 90, No. 1, 77-103 (1993; Zbl 0815.16006)]. For a two-peak poset \(S\), it was shown by \textit{S. Kasjan} and \textit{D. Simson} [in: Proc. sixth int. Conf. on Representations of Algebras, CMS Conf. Proc. 14, 245-284 (1993; Zbl 0834.16010) and J. Algebra 172, No. 2, 506-529 (1995; Zbl 0831.16010)] that if \(S\) is of tame prinjective type, then its Tits form is weakly non-negative (i.e. non-negative on the rational vectors with non-negative coordinates), and that the Tits form of \(S\) is weakly non-negative if and only if \(S\) contains no subset from a certain list of 47 one-peak and two-peak posets. In view of the well-known theorem for one-peak posets saying that the weak non-negativity of the Tits form is equivalent to the tameness of prinjective type, it is natural to ask whether the theorem is true for two-peak posets. The authors prove the theorem for a certain class of two-peak posets, which they call upper chain reducible posets. It follows that the upper chain reducible posets of tame prinjective type can be characterized both in terms of their Tits form and in terms of their subsets. However, the authors say that they have an example of a two-peak poset of wild prinjective type with a weakly non-negative Tits form, which shows that the above characterizations do not apply to all two-peak posets of tame prinjective type.
scientific article; zbMATH DE number 858649
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