The representations of nested composition algebras

In this paper, we investigate the basis graph of the monoid algebra of a submonoid of the monoid of mappings from N = {1,…,n} to itself, defined by a nested sequence of compositions of N. Each such monoid is a left regular band (LRB), that is, a semigroup S satisfying x² = x and xyx = xy for all x,y∈S. This class is su?ciently rich that every path algebra of an acyclic quiver can be embedded in such a monoid algebra. The multiplication in the monoid algebra has a particularly simple quasi-multiplicative form, allowing definition over the integers. Combining this with a formula for Ext-groups for LRBs due to Margolis et al. [6 Margolis, S., Saliola, F., Steinberg, B. (2015). Combinatorial topology and the global dimension of algebras arising in combinatorics. J. Eur. Math Soc. 17(12):3037–3080.[Crossref], [Web of Science ®] , [Google Scholar]], we get a simple criterion for the nested composition algebras to be hereditary.