Representations of finite posets and valuated groups
Let \(\mathfrak V\) denote the category of finite valuated \(p\)-groups and for a partially ordered set \((P,\leq)\) let \(\text{Rep}(P)\) be all sequences \((U,U(s), s\in P)\) of finite dimensional vector spaces over \(\mathbb{F}_ p\) with \(U(s)\subseteq U(t)\) whenever \(s \leq t\). \(\text{Rep}_ 0(P)\) are all those representations with \(U = \sum_{s\in P}U(s)\). There is a canonical functor \(H_ p:{\mathfrak V}\to\text{Rep}_ 0(P^{op})\), where \(P\) arises from the \(v\)-heights in a group \(G\) and \(P^{op}\) is \(P\) with the reverse order. In order to make applicable results from representations of partially ordered sets to valuated \(p\)-groups a correspondence \(\text{Rep}_ 0(P)\to{\mathfrak V}\) is needed. In general this will not be a full additive functorial embedding. In order to obtain a functor, the category \(\mathfrak V\) is changed into \(\mathfrak V/\mathfrak A\), which contains the same objects as \(\mathfrak V\) and morphisms are maps from \(\mathfrak V\) modulo those \(\varphi\) with \(\varphi(G[p]) = 0\). In this case \(H_ p\) is a functor \({\mathfrak V}/{\mathfrak A} \to \text{Rep}_ 0(P^{op})\). For a finite poset of finite valuated trees a mapping \(F_ p:\text{Rep}_ 0(P^{op})\to{\mathfrak V}/{\mathfrak A}\) is constructed, which preserves isomorphism and direct sums and which sends rank 1 representations to simply presented valuated groups. Moreover, if each tree in \(P\) is a pole, then \(F_ p\) is an additive functor on \({\mathfrak V}/{\mathfrak A}\). Those join-irreducible orderings of finite valuated trees with \(H_ pF_ p(U)\cong U\) for all \(U \in \text{Rep}_ 0(P^{op})\) are characterized. The image of \(F_ p\) are extensions of \(p\)-bounded groups by simply presented valuated groups. If \(P\) is a finite poset, direct sums of rank 1 representations can be described by invariants. It turns out that there is a correspondence between these invariants and Ulm invariants.
category of finite valuated \(p\)-groups
representations of partially ordered sets
scientific article; zbMATH DE number 168180