Bruce W. Westbury

Confluence theory for graphs

We develop a theory of confluence of graphs. We describe an algorithm for proving that a given system of reduction rules for abstract graphs and graphs in surfaces is locally confluent. We apply this algorithm to show that each simple Lie algebra of rank at most 2, gives rise to a confluent system of reduction rules of graphs (via Kuperbergs spiders) in an arbitrary surface. As a further consequence of this result, we find canonical bases of SU_3-skein modules of cylinders over orientable surfaces.

R-matrices and the magic square

The distinguished representations associated with the rows of the Freudenthal magic square have a uniform tensor product graph with edges labelled by linear functions of the dimension of the corresponding division algebra.

Invariant tensors and cellular categories

Let U be the quantised enveloping algebra associated to a Cartan matrix of finite type. Let W be the tensor product of a finite list of highest weight representations of U. Then End(U) (W) has a basis called the dual canonical basis and this gives an integral form for End(U) (W). We show that this integral form is cellular by using results due to Lusztig

Invariant tensors for the spin representation of

We give a graphical calculus for the invariant tensors of the eight dimensional spin representation of the quantum group Uq(B3). This leads to a finite confluent presentation of the centraliser algebras of the tensor powers of this representation and a construction of a cellular basis.

\frak{so}

Let V be the representation of the quantized enveloping algebra of

(7)

\mathfrak{gl}(n)

Web bases for the general linear groups

which is the q-analogue of the vector representation and let V∗ be the dual representation. We construct a basis for

Enumeration of non-positive planar trivalent graphs

\bigotimes^{r}(V \oplus V^{*})

Series of Nilpotent Orbits

with favorable properties similar to those of Lusztig’s dual canonical basis. In particular our basis is invariant under the bar involution and contains a basis for the subspace of invariant tensors.

An R-matrix for D(3)4

In this paper we construct inverse bijections between two sequences of finite sets. One sequence is defined by planar diagrams and the other by lattice walks. In [13] it is shown that the number of elements in these two sets are equal. This problem and the methods we use are motivated by the representation theory of the exceptional simple Lie algebra G2. However in this account we have emphasised the combinatorics.