The cyclic sieving phenomenon: a survey

The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and f(q) be a polynomial in q with nonnegative integer coefficients. Then the triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g, and w is a root of unity chosen to have the same order as g. It might seem improbable that substituting a root of unity into a polynomial with integer coefficients would have an enumerative meaning. But many instances of the cyclic sieving phenomenon have now been found. Furthermore, the proofs that this phenomenon hold often involve interesting and sometimes deep results from representation theory. We will survey the current literature on cyclic sieving, providing the necessary background about representations, Coxeter groups, and other algebraic aspects as needed.Comment: 48 pages, 3 figures, the sedcond version contains numerous changes suggested by colleagues and the referee. To appear in the London Mathematical Society Lecture Note Series. The third version has a few smaller change

Characteristic and Ehrhart polynomials

Let A be a subspace arrangement and let chi(A,t) be the characteristic polynomial of its intersection lattice L(A). We show that if the subspaces in A are taken from L(B_n), where B_n is the type B Weyl arrangement, then chi(A,t) counts a certain set of lattice points. One can use this result to study the partial factorization of chi(A,t) over the integers and the coefficients of its expansion in various bases for the polynomial ring R[t]. Next we prove that the characteristic polynomial of any Weyl hyperplane arrangement can be expressed in terms of an Ehrhart quasi-polynomial for its affine Weyl chamber. Note that our first result deals with all subspace arrangements embedded in B_n while the second deals with all finite Weyl groups but only their hyperplane arrangements.Comment: 16 pages, 1 figure, Latex, to be published in J. Alg. Combin. see related papers at http://www.math.msu.edu/~saga

Particles, environments and possible ecologies in the Jovian atmosphere

The eddy diffusion coefficient is estimated as a function of altitude, separately for the Jovian troposphere and mesosphere. Complex organic molecules produced by the Ly alpha photolysis of methane may possibly be the absorbers in the lower mesosphere which account for the low reflectivity of Jupiter in the near ultraviolet. The optical frequency chromophores are localized at or just below the Jovian tropopause. Candidate chromophore molecules must satisfy the condition that they are produced sufficiently rapidly that convective pyrolysis maintains the observed chromophore optical depth. The condition is satisfied if complex organic chromophores are produced with high quantum yield by NH3 photolysis at less than 2,300 A. Jovian photoautotrophs in the upper troposphere satisfy this condition well, even with fast circulation, assuming only biochemical properties of comparable terrestrial organisms. An organism in the form of a thin, gas filled balloon can grow fast enough to replicate if (1) it can survive at the low mesospheric temperatures, or if (2) photosynthesis occurs in the troposphere