Algebras associated with LSGOP
An LSGOP (lower substractive G orbit poset) for a finite group G is a rooted poset equipped with a G-action satisfying the following conditions: (1) the G-action on P is transitive; and (2) if \(\rho\) is the root of P, then x.\(\rho\leq y.\rho \leq z.\rho\) \(\Leftrightarrow\) \((x^{- 1}y.\rho \leq x^{-1}z.\rho\) and x.\(\rho\leq z.\rho)\) for all x, y, z in G. For the main part of the paper, the stabiliser of the root \(\rho\) is taken to be \(\{\) \(1\}\) and P is identified with G with the root \(\rho\) being identified with 1 of G. Thus G is a group with a partial order P and least element 1. If k is a field, the k-algebra \(k[G]_ P\) associated to with P is defined to be the k-vector space on symbols \(u_ g\) (g\(\in G)\) with a multiplication \(u_ g.u_ h=e_ P(g,h)u_{gh}\) (g,h\(\in G)\), where \(e_ P(g,h)\) is 1 (\(\in k)\) if \(g\leq gh\) and is 0 otherwise. Condition (2) is then equivalent to \(k[G]_ P\) being associative, i.e. the ``2- cosickle'' condition \(e_ P(g,h)e_ P(gh,k)=e_ P(h,k)e_ P(g,hk)\) holds. Such algebras and LSGOP's are defined and motivated in a paper by \textit{D. Haile}, \textit{R. Larson} and \textit{M. Sweedler} [Am. J. Math. 105, 689-814 (1983; Zbl 0523.13005)]. Section 2 shows that if P and Q are trees on G, then \(k[G]_ P\sim k[G]_ Q\) as k-algebras if and only if there is a poset isomorphism \(\sigma\) : \(P\to Q\) such that \(\sigma (g)\sigma (h)=\sigma (gh)\) holds whenever \(e_ P(g,h)=1.\) In section 3, various invariants of k-algebras \(k[G]_ P\) are obtained when P is not necessarily a tree. These include the numbers of elements at various levels above the root and also down from maximal elements (ages). An invariant diagram is constructed from \(k[G]_ P\) together with sources, sinks and isolated points. These have been sufficient to distinguish the different algebras \(k[G]_ P\) for all groups of order \(\leq 5\), by a case-by-case check. Section 4 gives a breakup of the radical of \(k[G]_ P\) as a direct sum of indecomposable right (or left) ideals, when P is a tree.
scientific article; zbMATH DE number 4063350