590 research outputs found
Cubic Harmonics and Bernoulli Numbers
The functions satisfying the mean value property for an n-dimensional cube
are determined explicitly. This problem is related to invariant theory for a
finite reflection group, especially to a system of invariant differential
equations. Solving this problem is reduced to showing that a certain set of
invariant polynomials forms an invariant basis. After establishing a certain
summation formula over Young diagrams, the latter problem is settled by
considering a recursion formula involving Bernoulli numbers.
Keywords: polyhedral harmonics; cube; reflection groups; invariant theory;
invariant differential equations; generating functions; partitions; Young
diagrams; Bernoulli numbers.Comment: 18 pages, 3 figure
Tropical cycles and Chow polytopes
The Chow polytope of an algebraic cycle in a torus depends only on its
tropicalisation. Generalising this, we associate a Chow polytope to any
abstract tropical variety in a tropicalised toric variety. Several significant
polyhedra associated to tropical varieties are special cases of our Chow
polytope. The Chow polytope of a tropical variety is given by a simple
combinatorial construction: its normal subdivision is the Minkowski sum of
and a reflected skeleton of the fan of the ambient toric variety.Comment: 22 pp, 3 figs. Added discussion of arbitrary ambient toric varieties;
several improvements suggested by Eric Katz; some rearrangemen
ADE surfaces and their moduli
We define a class of surfaces corresponding to the ADE root lattices and
construct compactifications of their moduli spaces as quotients of projective
varieties for Coxeter fans, generalizing Losev-Manin spaces of curves. We
exhibit modular families over these moduli spaces, which extend to families of
stable pairs over the compactifications. One simple application is a geometric
compactification of the moduli of rational elliptic surfaces that is a finite
quotient of a projective toric variety.Comment: A streamlined and expanded versio
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