The thesis of this paper is that considering the general setting of monoids admitting such a triangularity, namely J-trivial monoids, sheds further light on the topic. This is a step in an ongoing effort to use representation theory to automatically extract combinatorial structures from (monoid) algebras, often in the form of posets and lattices, both from a theoretical and computational point of view, and with an implementation in Sage.
Motivated by ongoing work on related monoids associated to Coxeter systems, and building on well-known results in the semi-group community (such as the description of the simple modules or the radical), we describe how most of the data associated to the representation theory (Cartan matrix, quiver) of the algebra of any J-trivial monoid M can be expressed combinatorially by counting appropriate elements in M itself. As a consequence, this data does not depend on the ground field and can be calculated in O(n2), if not O(nm), where n=|M| and m is the number of generators. Along the way, we construct a triangular decomposition of the identity into orthogonal idempotents, using the usual M\"obius inversion formula in the semi-simple quotient (a lattice), followed by an algorithmic lifting step.
Applying our results to the 0-Hecke algebra (in all finite types),
we recover previously known results and additionally provide an
explicit labeling of the edges of the quiver. We further explore
special classes of J-trivial monoids, and in particular monoids of
order preserving regressive functions on a poset, generalizing known
results on the monoids of nondecreasing parking functions.
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