Some problems related to plethysm

This thesis is concerned with plethysm. It investigates certain plethysm coefficients and also studies a diagram algebra whose representation theory is related to plethysm. We use techniques involving plethystic semistandard Young tableaux in order to provide information about near-maximal constituents of the plethysm sν ◦sµ when µ = (m),(12 ) or (2, 1). We study further the case where µ = (12 ) by the means of a recursive formula of Law and Okitani. We study the ramified partition algebra, proving some new results about its representation theory. We show that the ramified partition algebra is a cellular algebra and investigate its cell modules. We show that the cell modules of the ramified partition algebra form a stratifying system, and hence prove an analogue of the Hemmer-Nakano theorem for this algebra. We give partial results on the semisimplicity of the ramified partition algebra over C, making a conjecture for the general case. Finally, we study the restriction of the cell modules for the ramified partition algebra to the partition algebra, investigating two filtrations and making progress on the decomposition of such modules.