Here n ∈ N, k is a field, Sn is the symmetric group, δ ∈ k, and Pn(δ) is the partition algebra over k. Our aim in this note is to study the representation theory of a subalgebra P

In positive characteristic, the Specht modules S λ corresponding to partitions λ are not necessarily irreducible, and understanding their structure is vital to understanding the representation theory of the symmetric group. In this paper, we address the related problem of finding the spaces of homomorphisms between Specht modules. Indeed in [2], Carter and Payne showed that the space of homomorphisms from S λ to S µ is non-zero for certain pairs of partitions λ and µ where the Young diagram for µ is obtained from that for λ by moving several nodes from one row to another. We also consider partitions of this type, and, by explicitly examining certain combinations of semi-standard homomorphisms, we are able to give a constructive proof of the Carter-Payne theorem and to generalise it.

We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics. Classification (1991): 16G10, 20G05, 16S99, 17B10. Mathematics Subject

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