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

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