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Register a new userOn irreducible maps and slices
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Added by
J\\'er\\'emie Bouttier
Publication date
2013
fields
Physics
and research's language is
English
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We consider the problem of enumerating d-irreducible maps, i.e. planar maps whose all cycles have length at least d, and such that any cycle of length d is the boundary of a face of degree d. We develop two approaches in parallel: the natural approach via substitution where these maps are obtained from general maps by a replacement of all d-cycles by elementary faces, and a bijective approach via slice decomposition which consists in cutting the maps along shortest paths. Both lead to explicit expressions for the generating functions of d-irreducible maps with controlled face degrees, summarized in some elegant pointing formula. We provide an equivalent description of d-irreducible slices in terms of so-called d-oriented trees. We finally show that irreducible maps give rise to a hierarchy of discrete integrable equations which include equations encountered previously in the context of naturally embedded trees.
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We derive a formula for the generating function of d-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which we recover by taking d=0. As an application, we obtain an expression for the number of d-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges (d=2), 4-irreducible maps and maps of girth at least 6 (d=4). Our derivation is based on a tree interpretation of the various encountered generating functions.
Ordinary maps satisfy topological recursion for a certain spectral curve $(x, y)$. We solve a conjecture from arXiv:1710.07851 that claims that fully simple maps, which are maps with non self-intersecting disjoint boundaries, satisfy topological recursion for the exchanged spectral curve $(y, x)$, making use of the topological recursion for ciliated maps arXiv:2105.08035.
A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten calculus for matrix integrals. The goal of this paper is to present two independent proofs that are purely combinatorial and generalise in various directions, such as to the setting of stuffed maps and hypermaps. The main motivation to understand the relation between ordinary and fully simple maps is the fact that it could shed light on fundamental, yet still not well-understood, problems in free probability and topological recursion.
We determine the multiplicity of the irreducible representation V(n) of the simple Lie algebra sl(2,C) as a direct summand of its fourth exterior power $Lambda^4 V(n)$. The multiplicity is 1 (resp. 2) if and only if n = 4, 6 (resp. n = 8, 10). For these n we determine the multilinear polynomial identities of degree $le 7$ satisfied by the sl(2,C)-invariant alternating quaternary algebra structures obtained from the projections $Lambda^4 V(n) to V(n)$. We represent the polynomial identities as the nullspace of a large integer matrix and use computational linear algebra to find the canonical basis of the nullspace.
We study the undirected divisibility graph in which the vertex set is a finite subset of consecutive natural numbers up to N.We derive analytical expressions for measures of the graph like degree, clustering, geodesic distance and centrality in terms of the floor functions and the divisor functions. We discuss how these measures depend on the vertex labels and the size of graph N. We also present the specific case of prime vertices separately as corollaries. We could explain the patterns in the local measures for a finite size graph as well as the trends in global measures as the size of the graph increases.
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