Representation Theory of the Partition Algebras and Symmetric Groups

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BACKGROUND: REPRESENTATIONS OF THE SYMMETRIC GROUPSOne of the most important conceptual tools in the modern sciences is the notion of symmetry. A precise definition was given by the mathematician Hermann Weyl: symmetry is the invariance of an object ( e.g. a configuration of particles in space) under certain transformations. The set of all transformations leaving an object unchanged form what is called a group. Representation theory is the study of how an abstract group ( or an algebra) can transform a space. The symmetric groups form one of the most important families of groups, as any finite group can be found inside some symmetric group. Over the complex numbers the representations of the symmetric groups were classified and constructed by Young in 1930. However, over fields of positive characteristic ( the so-called modular case) the situation is considerably more complicated. A classification of the irreducible representations (that is the building blocks which, glued together, form any representation) exists but constructing these and understanding how they can be glued together are still major open problems in this area. A very profitable approach has been to relate the representations of the symmetric groups to that of other algebraic objects which admit richer mathematical structures. The classical example is the relationship with the general linear group via Schur-Weyl duality, which allows to transfer some of the tools from Lie theory to the symmetric group side. More recently, Jones [ll] and Martin [12] proved the existence of a similar duality, over the complex numbers, between the symmetric group and another algebraic structure which originally arose in Mathematical Physics, namely the partition algebra. Although this duality was established in the 1990's, surprisingly little work has been done since to develop and exploit this connection to advance our understanding of the representation theory of the symmetric group, especially in the modular case. MAIN OBJECTIVES OF THE PROPOSED RESEARCHIt is widely expected that the duality established by Jones and Martin will carry over to arbitrary fields, although as far as I am aware there is no available proof yet. My first objective is therefore to extend the duality between the symmetric group and the partition algebra to fields of positive characteristics. My second objective will then be to develop a systematic study of the functors between the two module categories which arise from this duality. I expect to obtain substantial new insight into the modular representation theory of the symmetric group in this way. It should be noted that many open problems also remain in the modular representation theory of the partition algebra. My third objective aims to address these by constructing an 'affine version' of the partition algebra. This is inspired by similar constructions for the symmetric group and the Brauer algebra which have brought very significant insights into their representation theory. METHODOLOGY AND WORK PLANTo reach my first objective (the extension of the duality to the modular case) I plan to use methods similar to those developed in invariant theory in [6] and for the Brauer algebra in [7]. The study of the so-called Schur functors arising from this duality (second objective) will most likely require the introduction of new classes of modules for the partition algebra, such as generalised permutation and Young modules (see [10] for analoguous results for the Brauer algebras). I also plan to use some of the methods developed in [9] in the classical Schur-Weyl duality. It should be noted, however, that our new setting has a major difference from the Brauer or the classical context in that we not have a Lie theory object in play. I expect that the third objective (the construction of an affine partition algebra), will provide the necessary extra structure in this case.