In 1979, Norton showed that the representation theory of the 0-Hecke algebra
admits a rich combinatorial description. Her constructions rely heavily on some
triangularity property of the product, but do not use explicitly that the
0-Hecke algebra is a monoid algebra.
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 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(n^2), 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.Comment: 41 pages; 4 figures; added Section 3.7.4 in version 2; incorporated
comments by referee in version