On the representation theory of finite Jtrivial monoids
Abstract
In 1979, Norton showed that the representation theory of the 0Hecke algebra admits a rich combinatorial description. Her constructions rely heavily on some triangularity property of the product, but do not use explicitly that the 0Hecke algebra is a monoid algebra. The thesis of this paper is that considering the general setting of monoids admitting such a triangularity, namely Jtrivial 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 wellknown results in the semigroup 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 Jtrivial 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 Möbius inversion formula in the semisimple quotient (a lattice), followed by an algorithmic lifting step. Applying our results to the 0Hecke 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 Jtrivial monoids, and in particular monoids of order preserving regressive functions on a poset, generalizing known results on the monoids of nondecreasing parking functions.
 Publication:

arXiv eprints
 Pub Date:
 October 2010
 arXiv:
 arXiv:1010.3455
 Bibcode:
 2010arXiv1010.3455D
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Combinatorics;
 20M30;
 16G99;
 20C08;
 06F05
 EPrint:
 41 pages