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1. CJM 2009 (vol 61 pp. 1300)
Monodromy Groups and Self-Invariance For every polytope $\mathcal{P}$ there is the universal regular
polytope of the same rank as $\mathcal{P}$ corresponding to the
Coxeter group $\mathcal{C} =[\infty, \dots, \infty]$. For a given
automorphism $d$ of $\mathcal{C}$, using monodromy groups, we
construct a combinatorial structure $\mathcal{P}^d$. When
$\mathcal{P}^d$ is a polytope isomorphic to $\mathcal{P}$ we say that
$\mathcal{P}$ is self-invariant with respect to $d$, or
$d$-invariant. We develop algebraic tools for investigating these
operations on polytopes, and in particular give a criterion on the
existence of a $d$\nobreakdash-auto\-morphism of a given order. As an application,
we analyze properties of self-dual edge-transitive polyhedra and
polyhedra with two flag-orbits. We investigate properties of medials
of such polyhedra. Furthermore, we give an example of a self-dual
equivelar polyhedron which contains no polarity (duality of order
2). We also extend the concept of Petrie dual to higher dimensions,
and we show how it can be dealt with using self-invariance.
Keywords:maps, abstract polytopes, self-duality, monodromy groups, medials of polyhedra Categories:51M20, 05C25, 05C10, 05C30, 52B70 |
2. CJM 1997 (vol 49 pp. 865)
Maps in locally orientable surfaces and integrals over real symmetric surfaces The genus series for maps is the generating series for the
number of rooted maps with a given number of vertices and
faces of each degree, and a given number of edges. It captures
topological information about surfaces, and appears in questions
arising in statistical mechanics, topology, group rings,
and certain aspects of free probability theory. An expression
has been given previously for the genus series for maps in
locally orientable surfaces in terms of zonal polynomials. The
purpose of this paper is to derive an integral representation
for the genus series. We then show how this can be used in
conjunction with integration techniques to determine the genus
series for monopoles in locally orientable surfaces. This
complements the analogous result for monopoles in orientable
surfaces previously obtained by Harer and Zagier. A conjecture,
subsequently proved by Okounkov, is given for the evaluation
of an expectation operator acting on the Jack symmetric function.
It specialises to known results for Schur functions and zonal
polynomials.
Categories:05C30, 05A15, 05E05, 15A52 |