Jucys-Murphy elements and the representations of partition algebras
by John Enyang
Abstract: Beginning with a presentation for the partition algebras given by Halverson and Ram, we provide a recursive definition for a large family of commuting (Jucys-Murphy) elements in the partition algebras, together with an integral Murphy-type basis for the partition algebras, with respect to which the commuting family acts triangularly. Consideration of the eigenvalues obtained by the triangular action of the Jucys-Murphy elements on the Murphy basis will show that this recursively defined commuting family of Jucys-Murphy elements coincides exactly with the family of commuting elements given diagrammatically by Halverson and Ram. Our presentation will conclude with a discussion of further applications of the Jucys-Murphy elements in the study of the representations of the partition algebras.
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