Mathew Pugh

Cardiff University | CU · School of Mathematics

Joint spectral measures associated to the rank two Lie group G2, including the representation graphs for the irreducible representations of G2 and its maximal torus, nimrep graphs associated to the G2 modular invariants have been studied. In this paper, we study the joint spectral measures for the McKay graphs (or representation graphs) of finite s...

In this paper we develop a taxonomy of errors which undergraduate mathematics students may make when tackling mathematical problems. We believe that a taxonomy would be useful for staff in giving feedback to students, and would facilitate students’ higher-level understanding of the types of errors that they could make.

The main goal of this paper is to classify $\ast$-module categories for the $SO(3)_{2m}$ modular tensor category. This is done by classifying $SO(3)_{2m}$ nimrep graphs and cell systems, and in the process we also classify the $SO(3)$ modular invariants. There are module categories of type $\mathcal{A}$, $\mathcal{E}$ and their conjugates. These co...

Spectral measures for fundamental representations of the rank two Lie groups A 2 , C 2 and G 2 have been studied. Since these groups have rank two, these spectral measures can be defined as measures over their maximal torus T 2 and are invari-ant under an action of the corresponding Weyl group, which are all subgroups of GL(2, Z). Here we consider...

Joint spectral measures associated to the rank two Lie group G2, including the representation graphs for the irreducible representations of G2 and its maximal torus, nimrep graphs associated to the G2 modular invariants have been studied. In this paper we study the joint spectral measures for the McKay graphs of finite subgroups of G2.

Joint spectral measures associated to the rank two Lie group $G_2$, including the representation graphs for the irreducible representations of $G_2$ and its maximal torus, nimrep graphs associated to the $G_2$ modular invariants have been studied. In this paper we study the joint spectral measures for the McKay graphs of finite subgroups of $G_2$.

Spectral measures for fundamental representations of the rank two Lie groups $A_2$, $C_2$ and $G_2$ have been studied. Since these groups have rank two, these spectral measures can be defined as measures over their maximal torus $\mathbb{T}^2$ and are invariant under an action of the corresponding Weyl group, which are all subgroups of $GL(2,\mathb...

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the compact connected rank two Lie group $B_2$ and its double cover the compact connected, simply-connected rank two Lie group $C_2$, including the McKay graphs for the irreducible representations of $C_2$ and...

We compute the Hochschild homology and cohomology, and cyclic homology, of almost Calabi-Yau algebras for SU(3) ADE graphs. These almost Calabi-Yau algebras are a higher rank analogue of the pre-projective algebras for Dynkin diagrams, which are SU(2)-related constructions. The Hochschild (co)homology and cyclic homology of A can be regarded as inv...

Braided subfactors of von Neumann algebras provide a framework for studying two dimensional conformal field theories and their modular invariants. We review this in the context of SU(3) conformal field theories through corresponding SU(3) braided subfactors and various subfactor invariants including spectral measures for the nimrep graphs, A_2-plan...

Generalizing Jones's notion of a planar algebra, we have previously introduced an A2-planar algebra capturing the structure contained in the double complex pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell system. We now introduce the notion of modules over an A2-planar algebra, and describe certain irreducible Hilbert A 2-...

We complete the computation of spectral measures for SU(3) nimrep graphs arising in subfactor theory, namely the SU(3) ADE{SU(3) \mathcal{ADE}} graphs associated with SU(3) modular invariants and the McKay graphs of finite subgroups of SU(3). For the SU(2) graphs the spectral measures distill onto very special subsets of the semicircle/circle, whil...

We determine the Nakayama automorphism of the almost Calabi-Yau algebra A associated to the braided subfactors or nimrep graphs associated to each SU(3) modular invariant. We use this to determine a resolution of A as an A-A bimodule, which will yield a projective resolution of A.

We determine spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with SU(3) modular invariants. Our methods also give an alternative approach to deriving the results of T. Banica and D. Bisch [Commun. Math. Phys. 269, No. 1, 259–281 (2007; Zbl 1122.46044)] for ADE graphs and subgroups of SU(2) and exp...

We determine the Nakayama permutation of the almost Calabi-Yau algebra associated to the braided subfactors or nimrep graphs associated to SU(3) modular invariants and the associated Hilbert series of dimensions.

We complete the realization by braided subfactors, announced by Ocneanu, of all SU(3)-modular invariant partition functions previously classified by Gannon.

We give a diagrammatic presentation of the A_2-Temperley-Lieb algebra. Generalizing Jones' notion of a planar algebra, we formulate an A_2-planar algebra motivated by Kuperberg's A_2-spider. This A_2-planar algebra contains a subfamily of vector spaces which will capture the double complex structure pertaining to the subfactor for a finite SU(3) AD...

We determine the cells, whose existence has been announced by Ocneanu, on all the candidate nimrep graphs except $\mathcal{E}_4^{(12)}$ proposed by di Francesco and Zuber for the SU(3) modular invariants classified by Gannon. This enables the Boltzmann weights to be computed for the corresponding integrable statistical mechanical models and provide...