2,343 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
The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory
The Jacobian conjecture is an old unsolved problem in mathematics, which has
been unsuccessfully attacked from many different angles. We add here another
point of view pertaining to the so called formal inverse approach, that of
perturbative quantum field theory.Comment: 22 pages, 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
The Higher Transvectants are Redundant
Let A, B denote generic binary forms, and let u_r = (A,B)_r denote their r-th
transvectant in the sense of classical invariant theory. In this paper we
classify all the quadratic syzygies between the u_r. As a consequence, we show
that each of the higher transvectants u_r, r>1, is redundant in the sense that
it can be completely recovered from u_0 and u_1. This result can be
geometrically interpreted in terms of the incomplete Segre imbedding. The
calculations rely upon the Cauchy exact sequence of SL_2-representations, and
the notion of a 9-j symbol from the quantum theory of angular momentum. We give
explicit computational examples for SL_3, g_2 and S_5 to show that this result
has possible analogues for other categories of representations.Comment: LaTeX, 38 page
The bipartite Brill-Gordan locus and angular momentum
This paper is a sequel to math.AG/0411110. Let P denote the projective space
of degree d forms in n+1 variables. Let e denote an integer < d/2, and consider
the subvariety X of forms which factor as L^{d-e} M^e for some linear forms
L,M. In the language of our earlier paper, this is the Brill-Gordan locus
associated to the partition (d-e,e).
In this paper we calculate the Castelnuovo regularity of X precisely, and
moreover show that X is r-normal for r at least 3. In the case of binary forms,
we give a classical invariant-theoretic description of the defining equations
of this locus in terms of covariants of d-ics. Modulo standard cohomological
arguments, the proof crucially relies upon showing that certain 3j-symbols from
the quantum theory of angular momentum are nonzero.Comment: LaTeX, 37 page
- …