Commuting families in Hecke and Temperley-Lieb algebras

Abstract We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group . We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.12

Jucys-Murphy elements and a presentation for partition algebras

We give a new presentation for the partition algebras. This presentation was discovered in the course of establishing an inductive formula for the partition algebra Jucys-Murphy elements defined by Halverson and Ram [European J. Combin. 26 (2005), 869-921]. Using Schur-Weyl duality we show that our recursive formula and the original definition of Jucys-Murphy elements given by Halverson and Ram are equivalent. The new presentation and inductive formula for the partition algebra Jucys-Murphy elements given in this paper are used to construct the seminormal representations for the partition algebras in a separate paper.Comment: 39 pages, 9 figures. Typos corrected and editorial changes made from v1-3. The final publication is available at springerlink.co

Letter of Appreciation: Peter Murphy Retires after a Lifetime of Dedication as Counsel to the Commandant of the Marine Corps

This letter reflects upon the retirement of Peter Murphy after 20 years of service as counsel to the commandant of the Marine Corps. Chief Judge Baker discusses Peter Murphy’s moral courage, common sense, and unflinching dignity while serving as counsel. He relates how Murphy has an abiding commitment to the great institutions of his life and of our lives: the rule of law, the military, and the Marine Corps

Room for God?

This essay begins with a review of several different approaches to the problem of divine action in the world, in light of the findings of science. Specifically, the speculation of Arthur Peacocke, John Polkinghorne and Nancey Murphy are elucidated and critiqued. Finding that Murphy, in founding her metaphysics on ontological reductionism, engenders theological reductionism, the author draws some conclusions are as to elements of a theologically attractive position