Abstract. Let u and v be permutations on n letters, with u ≤ v in Bruhat order. A Bruhat interval polytope Qu,v is the convex hull of all permutation vectors z = (z(1), z(2), . . . , z(n)) with u ≤ z ≤ v. Note that when u = e and v = w 0 are the shortest and longest elements of the symmetric group, Qe,w 0 is the classical permutohedron. Bruhat interval polytopes were studied recently in [KW13] by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety.In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality description and a dimension formula for Bruhat interval polytopes, and prove that every face of a Bruhat interval polytope is a Bruhat interval polytope. A key tool in the proof of the latter statement is a generalization of the well-known lifting property for Coxeter groups. Motivated by the relationship between the lifting property and R-polynomials, we also give a generalization of the standard recurrence for R-polynomials. Finally, we define a more general class of polytopes called Bruhat interval polytopes for G/P , which are moment map images of (closures of) totally positive cells in (G/P ) ≥0 , and are a special class of Coxeter matroid polytopes. Using tools from total positivity and the Gelfand-Serganova stratification, we show that the face of any Bruhat interval polytope for G/P is again a Bruhat interval polytope for G/P .

We define and study a variant of the center of mass of a polygon and, more generally, of a simplicial polytope which we call the Circumcenter of Mass (CCM). The Circumcenter of Mass is an affine combination of the circumcenters of the simplices in a triangulation of a polytope, weighted by their volumes. For an inscribed polytope, CCM coincides with the circumcenter.Our motivation comes from the study of completely integrable discrete dynamical systems, where the CCM is an invariant of the discrete bicycle (Darboux) transformation and of recuttings of polygons.We show that the CCM satisfies an analog of Archimedes' Lemma, a familiar property of the center of mass. We define and study a generalized Euler line associated to any simplicial polytope, extending the previously studied Euler line associated to the quadrilateral. We show that the generalized Euler line for polygons consists of all centers satisfying natural continuity and homogeneity assumptions and Archimedes' Lemma.Finally, we show that CCM can also be defined in the spherical and hyperbolic settings.

No abstract

Fourier-Dedekind sums are a generalization of Dedekind sums -important numbertheoretical objects that arise in many areas of mathematics, including lattice point enumeration, signature defects of manifolds and pseudo random number generators. A remarkable feature of Fourier-Dedekind sums is that they satisfy a reciprocity law called Rademacher reciprocity. In this paper, we study several aspects of Fourier-Dedekind sums: properties of general Fourier-Dedekind sums, extensions of the reciprocity law, average behavior of Fourier-Dedekind sums, and finally, extrema of 2dimensional Fourier-Dedekind sums.On properties of general Fourier-Dedekind sums we show that a general Fourier-Dedekind sum is simultaneously a convolution of simpler Fourier-Dedekind sums, and a linear combination of these with integer coefficients. We show that Fourier-Dedekind sums can be extended naturally to a group under convolution. We introduce "Reduced Fourier-Dedekind sums", which encapsulate the complexity of a Fourier-Dedekind sum, describe these in terms of generating functions, and give a geometric interpretation.Next, by finding interrelations among Fourier-Dedekind sums, we extend the range on which Rademacher reciprocity Theorem holds.We go on to study the average behavior of Fourier-Dedekind sums, showing that the average behavior of a Fourier-Dedekind sum is described concisely by a lowerdimensional, simpler Fourier-Dedekind sum.Finally, we focus our study on 2-dimensional Fourier-Dedekind sums. We find tight upper and lower bounds on these for a fixed t, estimates on the argmax and argmin, and bounds on the sum of their "reciprocals".