Total positivity criteria for partial flag varieties (Q411782)

A matrix with real entries is called totally positive if all its minors are positive. G. Lusztig extended this classical subject to the totally positive variety \(G_{>0}\) in an arbitrary reductive group \(G\) and the totally positive varieties \(\left( P \setminus G \right) _{>0}\) for any parabolic subgroup \(P\) of \(G\).NEWLINENEWLINEIn 2001, in order to study total positivity in algebraic groups and canonical bases in quantum groups, \textit{S. Fomin} and \textit{A. Zelevinsky} introduced the class of cluster algebras [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)]. \textit{C. Geiß}, \textit{B. Leclerc} and \textit{J. Schröer} have studied cluster algebras associated with Lie groups of type \(\mathbb{A}\), \(\mathbb{D}\), \(\mathbb{E}\), and have modelled them by categories of modules over the Gelfand-Ponomarev preprojective algebras \(\Lambda\) of the same type. They have shown in [Invent. Math. 165, No. 3, 589--632 (2006; Zbl 1167.16009)] that each reachable maximal rigid \(\Lambda\)-module can be thought of as a seed of a cluster algebra structure on \(\mathbb{C}[N]\), the coordinate ring of a maximal unipotent subgroup of \(G\). They also attached to each standard parabolic subgroup \(P\) of \(G\) a certain subcategory \(\mathcal{C}_P\) of \(\text{mod} \Lambda\) and showed that each reachable maximal rigid \(\Lambda\)-module in \(\mathcal{C}_P\) gives a seed for a cluster algebra structure on \(\mathbb{C}[N_P]\), the coordinate ring of the unipotent radical of \(P\).NEWLINENEWLINEIn [{\textit{C. Geiss}}, {\textit{B. Leclerc}} and {\textit{J. Schröer}}, EMS Series of Congress Reports, 253--283 (2008; Zbl 1203.16014)], they conjectured that each basic maximal rigid \(\Lambda\)-module in \(\mathcal{C}_P\) gives rise to a total positivity criterium for the partial flag variety \(P \setminus G\). In the paper under review, the author partially answers this conjecture and showed that every reachable basic maximal rigid module in \(\mathcal{C}_P\) gives rise to a total positivity criterium which leads to a (generally infinite) number of criteria.