Understanding idempotents in diagram semigroups

Abstract

The principal concern of this document is to develop and expose methodology for enumerating idempotents in certain semigroups of diagrams in the sense of [76]. These semigroups are known to be significant in the representation theory of associated algebras. In particular these algebras are shown in many cases to be semisimple, giving certain idempotents (and in particular those of the monoids of concern) a prominent role in understanding certain features of the representation theory in this situation. The results developed here are mostly theoretical in nature. We propose two viewpoints leading to some combinatorial understanding of the idempotents in the Motzkin (respectively Jones and partial Jones) monoid. In the first instance, we construct a cell complex, whose connected components partition the set of all idempotents into small, manageable chunks that can be analysed uniformly starting from those of particularly low rank. The structure of this complex captures some intricate combinatorics in the semigroup in a fairly simple, uniform way, and reduces our problem to finding and characterising idempotents of particularly low rank. The latter viewpoint takes us closer to pure combinatorics; a family of parameters attached to the elements of the monoids in question. These are examined in the context of ordinary generating functions, counting the elements with various parameter profiles. In particular, important algebraic features of Motzkin pictures, such as degree, rank, idempotency, and membership in the Jones and partial Jones monoids, can be tested against parameter profiles, reducing the problem of understanding all three to that of a parametric underiii iv standing of only the Motzkin monoid. We can then amalgamate these families of techniques into the development of fast linear-space algorithms for counting elements of various parameter profiles by examining certain “convex” elements. In particular, the general problem of enumeration by parameter profile is reduced greatly to enumerating convex elements by parameter profile. As a corollary to this study of convexity, we observe that the sequence of numbers of idempotents (in each semigroup) of some fixed rank-deficiency d = (n − r) is equal (apart from the first couple of values) to some polynomial of degree d; for particularly low rank-deficiency, we calculate these polynomials. Finally, we can show that the problem of understanding these idempotents in this way reduces to the classical open problem in combinatorics of counting meanders, witnessing the fact that significant progress on the former problem would necessitate some development of a better understanding of the latter.