Universal skein theory for finite depth subfactor planar algebras

We describe an explicit finite presentation for a finite depth subfactor planar algebra. We also show that such planar algebras are singly generated with the generator subject to finitely many relations.Comment: 13 pages, 10 figure

Planar algebras, cabling and the Drinfeld double

We produce an explicit embedding of the planar algebra of the Drinfeld double of a finite-dimensional, semisimple and cosemisimple Hopf algebra $H$ into the two-cabling of the planar algebra of the dual Hopf algebra $H^*$ and characterise the image.Comment: 17 pages, 27 figure

Lengths and multiplicities of integrally closed modules over a two-dimensional regular local ring

Let $(R,{\bf m})$ be a two-dimensional regular local ring with infinite residue field. We prove an analogue of the Hoskin-Deligne length formula for a finitely generated, torsion-free, integrally closed $R$-module $M$. As a consequence, we get a formula for the Buchsbaum-Rim multiplicity of $F/M$, where $F = M^{**}$.Comment: 15 page

A note on generators for finite depth subfactor planar algebras

We show that a subfactor planar algebra of finite depth $k$ is generated by a single $s$-box, for $s \leq min\{k+4,2k\}$.Comment: 6 page

Note on infinite iterated crossed products of Hopf algebras and the Drinfeld double

For a finite dimensional Hopf algebra we show that an associated natural inclusion of infinite crossed products is the crossed product by the Drinfeld double, and that this is a characterisation of the double.Comment: 9 page

On the genesis of a determinantal identity

We show how to derive a $3 \times 3$ determinantal identity in 12 indeterminates that gives an explicit version of a result of Mohan Kumar and Pavaman Murthy on completing unimodular rows.Comment: 4 page

From subfactor planar algebras to subfactors

We present a purely planar algebraic proof of the main result of a paper of Guionnet-Jones-Shlaykhtenko which constructs an extremal subfactor from a subfactor planar algebra whose standard invariant is given by that planar algebra.Comment: 22 pages, 25 figure

Hilbert functions of points on Schubert varieties in the Grassmannian

We prove a recent conjecture of Kreiman and Lakshmibai about Hilbert functions of points on Schubert varieties in Grassmannians.Comment: 35 pages, latex, 5 figure

Planar algebras and Kuperberg's 3-manifold invariant

We recapture Kuperberg's numerical invariant of 3-manifolds associated to a semisimple and cosemisimple Hopf algebra through a `planar algebra construction'. A result of possibly independent interest, used during the proof, which relates duality in planar graphs and Hopf algebras, is the subject of a final section.Comment: 19 pages, 9 figure

Picture invariants and the isomorphism problem for complex semisimple Lie algebras

Consider complex semisimple Lie algebras of a given dimension specified by their structure constants. We describe a finite collection of rational functions in the structure constants that form a complete set of invariants: two sets of structure constants determine isomorphic algebras if and only if each of these rational functions has the same value on the two sets. This collection is indexed by all chord diagrams with the number of chords being bounded by a function of the dimension.Comment: Latex, 11 pages, Uses packages epsfig and psfra