Semigroup theory at the cusp of representation theory and combinatorial algebra

Representation theory and combinatorial algebra reduce difficult or abstract questions, in the first case, to matrix arithmetic and, in the second case, to manipulations of symbols capable of algorithmic solution. Over four meetings, at least two or three themes will be explored, depending on the interests and background of the audience. The unifying discovery, by Graham and Lehrer in 1996, of cellularity of algebras grabbed world-wide attention because it provides templates for deciding in principle the main feature of the representation theory of a given algebra. There is a race to verify cellularity in as many examples as possible. In these talks the role of semigroups, twistings and anti-involutions will be explored to streamline the verification of cellularity in important cases and elucidate connections with classical representation theory. Vassiliev link invariants have recursive formulae where the singular braid monoid of Baez and Birman plays a seminal role. This monoid is one of a wide spectrum to which methods from combinatorial algebra may be applied. Corran invented a chain technique which has been applied recently by Antony to understand the Birman mapping, which Corran conjectures is injective for all Coxeter type (verified by Paris in type A). Corran's chainability condition is related to Dehornoy's word reversing and other techniques, whch yield fast algorithmic solutions to geometric problems involving braids. Lavers, East and FitzGerald have found new and novel presentations of braid monoids, and new representations using groups cosets and automorphisms of free groups. A sample of ideas and illustrations will be provided which may form the basis for further research. Relationships between idempotents are central to the study of semigroups and agebras. They form geometries, called biordered sets (invented by Nambooripad) on which semigroups and algebras act (often by induced forms of conjugation). Almost all known theory is for regular semigroups. Recent work by Jordan and Roberts goes much further and provides wide and interesting classes of fundamental semigroups which need not be regular, suggesting a basis for a general extension theory for semigroups. The relationship with the free semigroup on a biordered set is far from well understood. McElwee combines combinatorial and graph theoretic techniques to study maximal subgroups raising intriguing questions and conjectures about free subgroups and their ranks. REFERENCES: The FitzGerald Trick (and Lallement´s Lemma), manifested in various contexts: [1] D.G. FitzGerald, On inverses of products of idempotents in regular semigroups, J. Austral. Math. Soc. 13 (1972), 335-337 [2] G. Lallement, Congruences et equivalences de Green sur un demi-group regulier, C.R. Acad. Sci. Paris, Ser. A 262 (1966), 47-129 [3] T.E. Hall, On regular semigroups, J. Algebra, 24 (1973), 1-24 [4] F. 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