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6th Conference on Formal Power Series \& Algebraic Combinatorics\\
6eme Colloque S\'eries Formelles et Combinatoire Alg\'ebrique\\
FPSAC/SFCA '94\\
DIMACS, May 23-27, 1994\\
{\large\bf Reading List} \\
{\bf Algebraic and analytic approaches for the genus series
of 2-cell embeddings on orientable and nonorientable surfaces }\\
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D.M.Jackson
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This bibliography contains material that I have used, or have found useful,
in this study of the genus series for maps, and their
application. This area, with its associated areas and areas of application
are extensive, and I make no claims of exhaustivity for this reading list.
On the contrary, it is a personal selection, but it will help the
interested reader to track back to related material. The annotations indicate,
for the most part, the relationship to the main themes of my talk, and do
not attempt to summarise the paper as a whole. Also included are references to
 open questions,
when these touch on the subject matter of the cited paper.

It is inconvenient for the purposes of this list that some papers are awaiting
 publication. I have included them, nevertheless, since they will appear soon,
 and then may be helpful to the interested reader.

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\appendix
\section{Background from Physics}
\begin{thebibliography}{999}
%-----------------------------------------------------------------------
\bibitem{biz} {\sc D.Bessis, C.Itzykson and J.B.Zuber},
{\em Quantum field theory techniques in graphical enumeration},
Adv. Applied Math.,
{\bf1} (1980), 109-157.

\noindent This is the first paper I encountered on the connexion between graph
 like objects and
physics (a colleague was reviewing it for Math. Reviews).  It is a classic in
 the area, and has prompted much interest. I had been aware of it since the
 `80's,  but it was much later,
while on sabbatic leave, that I had the opportunity to study it. By that
time I had applied methods from group algebra to the {\em monopole}
problem, earlier solved by Harer and Zagier by analytic means.  Itzykson, in
 particular,
has written extensively about these matters in physics.

\bibitem{gra} {\sc D.J.Gross and A.A.Migdal,}
{\em A nonperturbative treatment of two dimensional quantum gravity,}
Nuclear Physics {\bf B~340} (1990), 333-365.

The introduction contains answers to several important questions about the
 physics.
For example, what is the double scaling limit, how can the area on the surface
 remain
finite in the limit, how is area represented combinatorially, how is
curvature measured combinatorially, what do physicists mean by
{\em triangulations} (in fact, they mean ``embeddings of graphs'', or something
close), where does the original integral that is combinatorialised come
from in the first instance?

\bibitem{tH} {\sc G.'t Hooft},
{\em A planar diagram theory for string interactions},
Nuclear Physics
{\bf B72} (1974), 461-473.

Contains an early formalism for planar diagrams and their relationship
to the main integral formulation of the partition function. My impression
is that this was the starting point for the later work on planar diagrams,
or was very close to it. The ideas in this paper were then developed
by others, and further mathematical apparatus was added to the theory.

\bibitem{id} {\sc C.Itzykson and J-M.Drouffe},
{\em ``Statistical Field Theory,''}
Vol. 2, Cambridge University Press, Cambridge, 1990.
Section 10.4 is the one for combinatorialists, and then inch forward and
backwards from that point. If one is not a physicist, it eventually becomes
foggy. For a combinatorialist, it shows where the Feynman diagrams come in.

\bibitem{meh} {\sc M.L.Mehta},
{\em ``Random Matrices,''}
Academic Press, London, 1967.

Gives some basic methods for dealing with some of the integrals
that occur. In fact, a great deal more is needed, but this is a good start.
It also deals with the real case and the quaternionic case.
The real case is needed for nonorientable surfaces.
One of the main ideas is the use of properties of Hermite polynomials
to carry out certain integrations. The complex case is associated with
orientable surfaces, the case upon which I will concentrate in the talk.
The real case seems to be harder since a extra symmetrisation associated
with the Jacobian (the absolute value of the Vandermonde) of the diagonalising
 transformation is needed.

\bibitem{npw}
{\em ``Statistical Mechanics of Membranes and Surfaces,''}
eds. D.Nelson, T.Piran, and S.Weinberg.
World Scientific, 1989, Singapore.

This is a very nice account of the application of maps and surfaces
to a variety of areas. It is a collection of papers on these topics.

\bibitem{ps} {\sc V.Periwal and D.Shevitz,}
{\em Exactly solvable unitary matrix models: multicritical potentials and
 correlates,}
Nuclear Physics {\bf B~344} (1990), 731-746.

This gives a good account of the details of solution of certain matrix models.
It has a nice introduction for people who are not familiar with the question.
The bibliography is also useful for further reading.  There is a huge amount of
 material on this
topic in the physics journals, and often the differences between papers
seem to be incremental, at least to me as someone unfamiliar with the
applications. I have selected this paper because it is readable and instructive.

\end{thebibliography}

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\section{Combinatorial papers}
\begin{thebibliography}{999}

\bibitem{ajv1} {\sc G.E.Andrews, D.M.Jackson and T.I.Visentin},
{\em A hypergeometric analysis of the genus series for a class of 2-cell
embeddings in orientable surfaces},
SIAM. J. Mathematical Analysis, (to appear). [CORR 92-09]

The integral representation delivers the genus series for subclasses of maps in
 a
convoluted form. The question arises whether the genus series can be
presented in a natural parity- and degree- respecting form.
Parity is forced by Euler's formula. A suitable basis, adapted to hypergeometric
 analysis, is used, and details of a study of the genus series for {\em
 monopoles} and {\em dipoles} with respect to this basis are given.

\bibitem{darq} {\sc D.Arqu\`{e}s},
{\em Relations fonctionelles et d\'{e}nombrement des cartes
point\'{e}es sur le tore},
J. Combinatorial Theory (A),
{\bf84} (1987) 253-274.

Uses extensions of Tutte-type decompositions to count maps on the torus. Details
 of the constructions on these surfaces
are given in full.

\bibitem{benc} {\sc E.A.Bender and E.R.Canfield},
{\em The asymptotic number of rooted maps on a surface},
J. Combinatorial Theory (A),
{\bf43} (1986) 244-257.

Gives a general form for the genus series for maps on orientable
surfaces. The approach is asymptotic, and makes heavy use of Tutte-type
 decompositions.
It would be valuable to have a proof of this result, for fixed genus, through
 the integral
representation.  This would add to our understanding of the genus series.
In the application of the Tutte-type decompositions, here and elsewhere, the
 genus is specified at the outset,
and a recursive step is taken to determine the exact enumeration for embeddings
 on this
surface. The genus series is the sum of all of these contribution, marked by the
 genus
variable. That is to say, the genus series is constructed term by term, as a
 series
in the genus variable, by this process.

\bibitem{bcr3} {\sc E.A.Bender, E.R.Canfield and R.W.Robinson},
{\em The enumeration of maps on the torus and the projective plane},
Canad. Math. Bull,
{\bf31} (1988), 257-271.

Gives exact enumeration on the torus, double torus and the projective plane
by the use of Tutte-type decompositions. Full details of the constructions
are given.

\bibitem{fh} {\sc H.K.Farahat and G.Higman},
{\em The centres of symmetric group rings},
Proc. Roy. Soc., A
{\bf250} (1959), 212-221.

Has a bearing on the question of constructing a set of symmetric functions
which have the same connexion coefficients under multiplication as those of the
class algebra of the symmetric group. This is closely associated with the
determination of the genus series. A specific type of connexion coefficient is
 determined, and this was later encountered by Macdonald in an algebraic
 construction.

\bibitem{zg} {\sc Z.Gao},
{\em The number of degree restricted maps on general surfaces,}
(preprint) March 1993.

This uses extensions of Tutte-type decompositions to surfaces of higher genus
to obtain expressions for the general form of the genus series for Eulerian maps
(every vertex has even degree) for the orientable case and the nonorientable
 case.
Restrictions can be placed on degrees.

\bibitem{book} {\sc I.P.Goulden and D.M.Jackson},
{\em ``Combinatorial Enumeration,''}
Wiley Interscience,
New York, 1983.

Contains, for these purposes, a compact account of Tutte's approach for the
enumeration of maps on the sphere, a commentary on the "quadratic method", and
futher references to Tutte's work.  This approach has been extended in several
 ways by Arqu\`{e}s, Bender, Canfield, Gao, Walsh and others.

\bibitem{ccind} {\sc I.P.Goulden and D.M.Jackson},
{\em Combinatorial constructions for integrals over normally distributed random
 matrices},
Proc. Amer. Math. Soc, (to appear). [CORR 93-04].

Gives combinatorial constructions to bridge the gap described in~\cite{hss}. At
 one level,
the approach can be viewed as an application of the embedding theorem for maps,
 and this was
part of the original insight. In fact, the construction we give is made
 independent
of the surface and is stated as a factorisation analogous to the disjoint cycle
 decomposition.
In this sense I regard this as an application of ideas associated with maps.

\bibitem{mconj} {\sc I.P.Goulden and D.M.Jackson},
{\em Symmetric functions and Macdonald's result for top connexion coefficients
 in the symmetric
group}, J. Algebra, (to appear). [CORR 92-09].

Macdonald (private communication)  has an algebraic construction which
 reproduced the result
that appears in~\cite{fh}. We give a combinatorial construction which motivates
Macdonald's and is based on the embedding of 2-coloured trees. Enumerative
 properties of the symmetric functions are also given. Through other classes of
 maps (eg. unicursal maps)
 some further coefficients can be obtained. Unfortunately, it does not seem
possible to generalise this beyond these elementary classes.  Murphy elements
 provide a
way for carrying out symbolic computations to produce the symmetric functions.
However, no information beyond this is available. My belief is that progress on
 this
general question will lead to further information about maps.

\bibitem{GT} {\sc J.L.Gross and T.W.Tucker},
{\em ``Topological Graph Theory''},
Wiley Interscience, New York, 1987.

\bibitem{hss} {\sc P.J.Hanlon, R.P.Stanley, and J.R.Stembridge},
{\em Some combinatorial aspects of the spectra of normally distributed random
matrices},
Contemporary Math.,
{\bf138} (1992), 151-174.

Contains an account of some integrals giving expectations over certain
classes of random matrices. These results are then expressed in combinatorial
terms, paving the way for a combinatorial derivation of the original
expections. Deals with the complex case and with the real case.

\bibitem{hz} {\sc J.Harer and D.Zagier},
{\em The Euler characteristic of the moduli space of curves},
Invent. Math,
{\bf 85} (1986), 457-485.

The part of this paper which is associated with genus series is their
 determination of
the genus series for maps. Their approach is an analytic one, and gives a
 recurrence
equation for the genus series for {\em monopoles} (maps with exactly one
 vertex). This
is a three term recurrence equation, with no elementary combinatorial
 explanation at the
present.

\bibitem{dmj1} {\sc D.M.Jackson},
{\em Counting cycles in permutations by group characters, with an application
to a topological problem},
Trans. Amer. Math. Soc.,
{\bf299} (1987), 785-801.

This contains a group algebra approach to the problem raised in part of the
Harer and Zagier paper. The paper in fact deals with {\em monopoles}, as the
 title
very indirectly suggests. Subsequently, Visentin and I extended the algebraic
ideas to all maps. Zagier has asked for a combinatorial proof of the three term
recurrence that the coefficients of the genus series satisfy. There is an
inclination to believe that all 3-term recurrence equations are easily
 explainable,
combinatorially. This one does not seem to be, and having such an explanation
may help with other bijective questions mentioned here.

\bibitem{longmap} {\sc D.M.Jackson},
{\em On an integral representation for the genus series for 2-cell embeddings,}
Trans. Amer. Math. Soc., (to appear). [CORR 92-30]

Contains a derivation of the integral representation for orientable surfaces
by considering rotation systems and the ``embedding theorem'', and the idea of
discontinous integral representations of combinatorial functions. It also
 contains
an explicit determination of the genus series for {\em dipoles} (maps with
 exactly two
vertices, these having the same degree). It would be interesting to use such an
approach to derive for nonorientable surfaces a correspondence similar to the
 one
given in~\cite{JVone} for orientable surfaces.  Also contains the basic elements
 needed
for the extension to $k$-poles, although you have to read it carefully to see
 this.

\bibitem{dmjmap} {\sc D.M.Jackson},
{\em The genus series for maps}, [CORR 92-39].

Contains a purely symmetric function proof of the integral representation
of the genus series, with the aid of the embedding theorem. This avoids the
use of Haar measure and unitary diagonalisation argument for Hermitian complex
 matrices.

\bibitem{JVone} {\sc D.M.Jackson and T.I.Visentin},
{\em A character theoretic approach to embeddings of rooted maps in an
 orientable
surface of given genus},
Trans. Amer. Math. Soc.,
{\bf322} (1990), 343-363.

Gives the character theoretic formulation of the genus
series for general maps. It uses the "embedding theorem" in conjunction
with the group algebra of the symmetric group and a factorisation of
characters of irreducible representations of the symmetric group
to obtain a simple relationship between the genus series
for quadrangulations and all maps. No proof of this
is known by operations on surfaces. For the sphere, it corresponds to
a construction of Tutte's. However, Tutte's construction does not
generalise to all maps on all surfaces. There is no proof of this
relationship directly from the integral representation. It would be interesting
if this could be done, since it might help with finding the corresponding
result for nonorientable surfaces.

\bibitem{JVtwo} {\sc D.M.Jackson and T.I.Visentin},
{\em Character theory and rooted maps in an orientable surface of given genus:
face coloured maps},
Trans. Amer. Math.Soc.,
{\bf322} (1990), 365-376.

Extends the above to hypermaps ({\em i.e.} face 2-colourable maps).
Relationships between other classes of maps, such as triangulations, are also
 obtained.

\bibitem{JVregmaps} {\sc D.M.Jackson and T.I.Visentin},
{\em The genus series for regular maps}. [CORR 93-18].

Regular maps are maps whose vertices have the same degree. It is possible to use
 the integral
representation and to carry out the integration combinatorially by making use of
 combinatorial
properties of Hermite polynomials. The genus series for quadrangulations is
 included in the genus
series for regular maps. It is hoped that this representation of the genus
 series may lead to
an alternative attack on the $\phi^4$-model, since this is related to the genus
 series
for quadrangulations. The genus series for monopoles and dipoles are recoverable
 from it.

\bibitem{kluyver} {\sc J.C.Kluyver},
{\em A local probability problem},
Proc. Section of Sci., K. Akad. van Wet. te Amsterdam,
{\bf 8} (1906), 341-350.

Uses Bessel functions in a discontinuous integral representation of $<,$
in connexion with a probabilistic question. I have included this perhaps obscure
reference since it was, for me, the key to the crucial link between a
 combinatorial
sum that arises in the genus series and its integral representation.

\bibitem{macd} {\sc I.G.Macdonald},
{\em ``Symmetric functions and Hall polynomials,''}
Clarendon Press, Oxford, 1979.

This is the handbook on symmetric functions.

\bibitem{penn} {\sc R.C.Penner},
{\em Perturbative series and the moduli space of Riemann surfaces},
J.Differential Geometry,
{\bf27} (1988), 35-53.

This paper contains the details of a very ingenious use of Laguerre polynomials
to determine a series closely related to the genus series.

\bibitem{se} {\sc J-P.Serre},
{\em ``Linear Representations of Finite Groups,''}
 Springer-Verlag, New York, 1977.

\bibitem{tucol}
{\em ``Selected Papers of W.T.Tutte,''}
Vols. I and II, Eds. D.McCarthy and R.G.Stanton,
The Charles Babbage Research Centre, St. Pierre, Manitoba, 1979.

In particular, see the ``census'' papers for details of applications of the
 quadratic method,
and various deconpositions.

\end{thebibliography}
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