A model for statistical ranking is a family of probability distributions whose states are orderings of a fixed finite set of items. We represent the orderings as maximal chains in a graded poset. The most widely used ranking models are parameterized by rational function in the model parameters, so they define algebraic varieties. We study these varieties from the perspective of combinatorial commutative algebra. One of our models, the Plackett-Luce model, is non-toric. Five others are toric: the Birkhoff model, the ascending model, the Csiszár model, the inversion model, and the Bradley-Terry model. For these models we examine the toric algebra, its lattice polytope, and its Markov basis.

Abstract. For a finite real reflection group W and a W-orbit O of flats in its reflection arrangement – or equivalently a conjugacy class of its parabolic subgroups – we introduce a statistic noninvO(w) on w in W that counts the number of “O-noninversions ” of w. This generalizes the classical (non-)inversion statistic for permutations w in the symmetric group Sn. We then study the operator νO of right-multiplication within the group algebra CW by the element that has noninvO(w) as its coefficient on w. We reinterpret νO geometrically in terms of the arrangement of reflecting hyperplanes for W, and more generally, for any real arrangement of linear hyperplanes. At this level of generality, one finds that, after appropriate scaling, νO corresponds to a Markov chain on the chambers of the arrangement. We show that νO is self-adjoint and positive semidefinite, via two explicit factorizations into a symmetrized form pitpi. In one such factorization, the matrix pi is a generalization of the projection of a simplex onto the linear ordering polytope from the theory of social choice. In the other factorization of νO as pitpi, the matrix pi is the transition matrix for one of the well-studied Bidigare-Hanlon-Rockmore random walks on the chambers of an ar-