2,558 research outputs found
Feynman Diagrams in Algebraic Combinatorics
We show, in great detail, how the perturbative tools of quantum field theory
allow one to rigorously obtain: a ``categorified'' Faa di Bruno type formula
for multiple composition, an explicit formula for reversion and a proof of
Lagrange-Good inversion, all in the setting of multivariable power series. We
took great pains to offer a self-contained presentation that, we hope, will
provide any mathematician who wishes, an easy access to the wonderland of
quantum field theory.Comment: 13 diagram
Grassmann-Berezin Calculus and Theorems of the Matrix-Tree Type
We prove two generalizations of the matrix-tree theorem. The first one, a
result essentially due to Moon for which we provide a new proof, extends the
``all minors'' matrix-tree theorem to the ``massive'' case where no condition
on row or column sums is imposed. The second generalization, which is new,
extends the recently discovered Pfaffian-tree theorem of Masbaum and Vaintrob
into a ``Hyperpfaffian-cactus'' theorem. Our methods are noninductive, explicit
and make critical use of Grassmann-Berezin calculus that was developed for the
needs of modern theoretical physics.Comment: 23 pages, 2 figures, 3 references adde
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