14,129 research outputs found
On the orbits of the product of two permutations
AbstractWe consider the following problem: given three partitions A,B,C of a finite set Ω, do there exist two permutations α and β such that A,B,C are induced by α, β and αβ respectively? This problem is NP-complete. However it turns out that it can be solved by a polynomial time algorithm when some relations between the number of classes of A,B,C hold
Multiple Flag Varieties of Finite Type
We classify all products of flag varieties with finitely many orbits under
the diagonal action of the general linear group. We also classify the orbits in
each case and construct explicit representatives. This generalizes the
classical Schubert decompostion, which states that the GL(n)-orbits on a
product of two flag varieties correspond to permutations. Our main tool is the
theory of quiver representations.Comment: 18pp. to appear in Adv. Mat
Maps, immersions and permutations
We consider the problem of counting and of listing topologically inequivalent
"planar" {4-valent} maps with a single component and a given number n of
vertices. This enables us to count and to tabulate immersions of a circle in a
sphere (spherical curves), extending results by Arnold and followers. Different
options where the circle and/or the sphere are/is oriented are considered in
turn, following Arnold's classification of the different types of symmetries.
We also consider the case of bicolourable and bicoloured maps or immersions,
where faces are bicoloured. Our method extends to immersions of a circle in a
higher genus Riemann surface. There the bicolourability is no longer automatic
and has to be assumed. We thus have two separate countings in non zero genus,
that of bicolourable maps and that of general maps. We use a classical method
of encoding maps in terms of permutations, on which the constraints of
"one-componentness" and of a given genus may be applied. Depending on the
orientation issue and on the bicolourability assumption, permutations for a map
with n vertices live in S(4n) or in S(2n). In a nutshell, our method reduces to
the counting (or listing) of orbits of certain subset of S(4n) (resp. S(2n))
under the action of the centralizer of a certain element of S(4n) (resp.
S(2n)). This is achieved either by appealing to a formula by Frobenius or by a
direct enumeration of these orbits. Applications to knot theory are briefly
mentioned.Comment: 46 pages, 18 figures, 9 tables. Version 2: added precisions on the
notion used for the equivalence of immersed curves, new references. Version
3: Corrected typos, one array in Appendix B1 was duplicated by mistake, the
position of tables and the order of the final sections have been modified,
results unchanged. To be published in the Journal of Knot Theory and Its
Ramification
Periodic-Orbit Theory of Universality in Quantum Chaos
We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory,
that full classical chaos is paralleled by quantum energy spectra with
universal spectral statistics, in agreement with random-matrix theory. For
dynamics from all three Wigner-Dyson symmetry classes, we calculate the
small-time spectral form factor as power series in the time .
Each term of that series is provided by specific families of pairs of
periodic orbits. The contributing pairs are classified in terms of close
self-encounters in phase space. The frequency of occurrence of self-encounters
is calculated by invoking ergodicity. Combinatorial rules for building pairs
involve non-trivial properties of permutations. We show our series to be
equivalent to perturbative implementations of the non-linear sigma models for
the Wigner-Dyson ensembles of random matrices and for disordered systems; our
families of orbit pairs are one-to-one with Feynman diagrams known from the
sigma model.Comment: 31 pages, 17 figure
Strings from Feynman Graph counting : without large N
A well-known connection between n strings winding around a circle and
permutations of n objects plays a fundamental role in the string theory of
large N two dimensional Yang Mills theory and elsewhere in topological and
physical string theories. Basic questions in the enumeration of Feynman graphs
can be expressed elegantly in terms of permutation groups. We show that these
permutation techniques for Feynman graph enumeration, along with the Burnside
counting lemma, lead to equalities between counting problems of Feynman graphs
in scalar field theories and Quantum Electrodynamics with the counting of
amplitudes in a string theory with torus or cylinder target space. This string
theory arises in the large N expansion of two dimensional Yang Mills and is
closely related to lattice gauge theory with S_n gauge group. We collect and
extend results on generating functions for Feynman graph counting, which
connect directly with the string picture. We propose that the connection
between string combinatorics and permutations has implications for QFT-string
dualities, beyond the framework of large N gauge theory.Comment: 55 pages + 10 pages Appendices, 23 figures ; version 2 - typos
correcte
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