arXiv subjects

Abdelmalek Abdesselam

Publications and source records attributed to Abdelmalek Abdesselam.

41 records · Page 3Linked to original sources

arXiv2005

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.

math.AG↗

arXiv2004

Brill-Gordan Loci, Transvectants and an Analogue of the Foulkes Conjecture

Combining a selection of tools from modern algebraic geometry, representation theory, the classical invariant theory of binary forms, together with explicit calculations with hypergeometric series and Feynman diagrams, we obtain the following interrelated results. A Castelnuovo-Mumford regularity bound and a projective normality result for the locus of hypersufaces that are equally supported on two hyperplanes. The surjectivity of an equivariant map between two plethystic compositions of symmetric powers; a statement which is reminiscent of the Foulkes-Howe conjecture. The nonvanishing of even transvectants of exact powers of generic binary forms. The nonvanishing of a collection of symmetric functions defined by sums over magic squares and transportation matrices with nonnegative integer entries. An explicit set of generators, in degree three, for the ideal of the coincident root locus of binary forms with only two roots of equal multiplicity.

math.AG↗

arXiv2004

A regularity result for a locus of Brill type

Let n,d be positive integers, with d even (say d=2e). Let X_(n,d) denote the locus of degree d hypersurfaces in P^n which consist of two e-fold hyperplanes. We bound the regularity of the ideal of this variety. Moreover, we show that this variety is r-normal for r at least 2. The proof of the latter part is is a result of a tripartite collaboration of algebraic geometry, classical invariant theory and theoretical physics. It is executed by reducing the question to a combinatorial calculation involving Feynman diagrams and hypergeometric functions.

math.AG↗

arXiv2003

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.

math.CO↗

arXiv2002

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.

math.CO↗