Moser et al. provide a formalization of heresthetics, the “art of political strategy”, in collective choice settings. In doing so they introduce the heresthetically stable set as the set of outcomes least susceptible to manipulation of issue dimension. In this note we correct a small error in the original paper, and close several open questions asked there in. We examine the heresthetically stable set as a tournament solution, establishing some basic properties it possesses, and many it does not posses. In addition, we relate the heresthetically stable set to other tournament solutions, notably the weak uncovered and refinements thereof. We find lack of vulnerability to heresthetic manipulation is contrary to many desirable properties of choice functions, notably majoritarian support.

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The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any upper triangular matrix. These subspaces are called ad-nilpotent ideals and there are Catalan number of such subspaces. Each ad-nilpotent ideal I meets a unique largest nilpotent orbit O I in the Lie algebra of all matrices. The main result of the paper is that under an equivalence relation on ad-nilpotent ideals studied by Mizuno and others, the equivalence classes are the ad-nilpotent ideals I such that O I = O for a fixed nilpotent orbit O. We include two applications of the result, one to the higher vanishing of cohomology groups of vector bundles on the flag variety and another to the Kazhdan-Lusztig cells in the affine Weyl group of the symmetric group. Finally, some combinatorial results are discussed.

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