Coxeter submodular functions and deformations of Coxeter permutahedra

We describe the cone of deformations of a Coxeter permutahedron, or equivalently, the nef cone of the toric variety associated to a Coxeter complex. This family of polytopes contains polyhedral models for the Coxeter-theoretic analogs of compositions, graphs, matroids, posets, and associahedra. Our description extends the known correspondence between generalized permutahedra, polymatroids, and submodular functions to any finite reflection group.Comment: Minor edits. To appear in Advances of Mathematic

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Maxwell equations and the redundant gauge degree of freedom

On transformation to the Fourier space $({\bf k}, \omega)$, the partial differential Maxwell equations simplify to algebraic equations, and the Helmholtz theorem of vector calculus reduces to vector algebraic projections. Maxwell equations and their solutions can then be separated readily into longitudinal and transverse components relative to the direction of the wave vector {\bf k}. The concepts of wave motion, causality, scalar and vector potentials and their gauge transformations in vacuum and in materials can also be discussed from an elementary perspective. In particular, the excessive freedom of choice associated with the gauge dependence of the scalar and the longitudinal vector potentials stands out with clarity in Fourier spaces. Since these potentials are introduced to represent the instantaneous longitudinal electric field, the actual cancellation in the latter of causal contributions arising from these potentials separately in most velocity gauges becomes an important issue. This cancellation is explicitly demonstrated both in the Fourier space, and for pedagogical reasons again in space-time. The physical origin of the gauge degree of freedom in the masslessness of the photon, the quantum of electromagnetic wave, is elucidated with the help of special relativity and quantum mechanics.Comment: 16 page