Abstract
In the context of the Eskin-Okounkov approach to the calculation of the volumes of the different strata of the moduli space of quadratic differentials, the important ingredients are the pillowcase weight probability distribution on the space of Young diagrams, and the asymptotic study of characters of permutations that near-involutions. In this paper we present various new results for these objects. Our results give light to unforeseen difficulties in the general solution to the problem, and they simplify some of the previous proofs.
Highlights
A quadratic differential on a Riemann surface S is a function defined on the tangent bundle ω : T S → C that locally looks like ω(z) = f (z)(dz)2, for some meromorphic function f with at most simple poles
We will say that two quadratic differentials ω and η defined on surfaces S and T are equivalent if there is a holomorphic diffeomorphism φ : S → T such that φ∗η = ω
The Pillowcase Distribution and Near-Involutions and poles. We encode this information in a partition ν = (ν1 ≥ ν2 ≥ · · · ≥ νk ≥ 1), in which a simple pole will be represented by a part νi equal to 1, a marked point will be a 2, and a zero of degree n will correspond to a part νi = n − 2
Summary
|Cη| is the size of the conjugacy class of the symmetric group corresponding to permutations whose cycle type is given by the partition η, and χλ(η) is the character of one such permutation in the irreducible representation of the symmetric group corresponding to the partition λ In practical terms, this means that in order to determine the volume of one of these strata one needs to compute the first few coefficients of the series (1.2), from there deduce what polynomial in the Eisenstein series it corresponds to, and make use of the quasimodularity to determine the leading term in the asymptotics. That there is no central limit theorem for the coupled pillowcase weights (1.3) as q → 1, and that there is no Wick-type theorem associated to the distribution they induce This paper elaborates on some of the results of the author’s PhD thesis [20], where a more detailed account of most of the proofs can be found
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