# School of Mathematics

## Prof Paul P Martin

Professor of Pure Mathematics

Head of Pure Mathematics

### Contact details

Room: 8.17

Tel: +44 (0)113 3437787

Email: P.P.Martin @ leeds.ac.uk

### Keywords

Representation theory

Diagram categories and algebras

Statistical Mechanics

## Research interests

Physics can be thought of an attempt to understand, and make use of, the physical world. Part of the process of understanding any system is to make models of it. Such models can range from scaled-down versions, to mathematical simulations. My research is concerned with making useful mathematical models.

Modeling as an approach to understanding is not limited to physics. One can attempt to model abstract structures (such as groups of symmetries) by modeling. In this setting the process is called Representation Theory.

We often find ourselves in the situation that a mathematical model of a physical system is still rather complicated, and benefits from modeling itself. Thus one aspect of my research is concerned with the representation theory of structures which in turn model phenomena in physics (typically in Statistical Mechanics).

A wonderful feature of this modeling chain is that the non-abstract end system is amenable to other approaches, such as experiment and physical intuition. Thus not only does the representation theory inform the physics, but also the physics informs the representation theory. By pursuing this line, many exciting new pieces of mathematics have been discovered.

In summary then, my research is concerned with the representation theory of structures (such as groups and algebras) used in physical modeling.

See the links on my personal homepage for many more details.

### Useful links

## Current postgraduate students

Amani Alfadhli (2014)

Eman Alluqmani (2014)

Alexander Bullivant (2013)

Ahmed Hussein (2012)

Konstantinos Meichanetzidis (2013)

## Publications

**Kádár Z, Martin P, Rowell E, Wang Z** LOCAL REPRESENTATIONS of the LOOP BRAID GROUP *Glasgow Mathematical Journal*, **59**, 359-378, 2017

DOI:10.1017/S0017089516000215

View abstract

**Bullivant A, Calçada M, Kádár Z, Martin P, Martins JF** Topological phases from higher gauge symmetry in 3+1 dimensions *Physical Review B*, **95**, 2017

DOI:10.1103/PhysRevB.95.155118

View abstract

**De Visscher M, Martin PP** On Brauer algebra simple modules over the complex field *Transactions of the American Mathematical Society*, **369**, 1579-1609, 2017

DOI:10.1090/tran/6716

View abstract

**Martin PP** The decomposition matrices of the Brauer algebra over the complex field *Transactions of the American Mathematical Society*, **367**, 1797-1825, 2014

DOI:10.1090/S0002-9947-2014-06163-1

View abstract

**Martin P, Mazorchuk V** On the representation theory of partial brauer algebras *Quarterly Journal of Mathematics*, **65**, 225-247, 2014

DOI:10.1093/qmath/has043

View abstract

**Martin P, Mazorchuk V** Partitioned binary relations *Mathematica Scandinavica*, **113**, 30-52, 2013

View abstract

**Marsh RJ, Martin P** Tiling bijections between paths and Brauer diagrams *J ALGEBR COMB*, **33**, 427-453, 2011

DOI:10.1007/s10801-010-0252-6

**Cox A, De Visscher M, Martin P** Alcove geometry and a translation principle for the Brauer algebra *J PURE APPL ALGEBRA*, **215**, 335-367, 2011

DOI:10.1016/j.jpaa.2010.04.023

**Cox A, De Visscher M, Martin P** A geometric characterisation of the blocks of the Brauer algebra *J LOND MATH SOC*, **80**, 471-494, 2009

DOI:10.1112/jlms/jdp039

**Cox A, de Visscher M, Martin P** The blocks of the brauer algebra in characteristic zero *Representation Theory*, **13**, 272-308, 2009

DOI:10.1090/S1088-4165-09-00305-7

View abstract

**Cox A, De Visscher M, Martin P** The blocks of the Brauer algebra in characteristic zero *Representation Theory. An Electronic Journal of the American Mathematical Society*, **13**, 272-308, 2009

DOI:10.1090/S1088-4165-09-00305-7

**Cox A, De Visscher M, Doty S, Martin P** On the blocks of the walled Brauer algebra *J ALGEBRA*, **320**, 169-212, 2008

DOI:10.1016/j.jalgebra.2008.01.026

**Martin P ** *On diagram categories, representation theory and statistical mechanics*NONCOMMUTATIVE RINGS, GROUP RINGS, DIAGRAM ALGEBRAS AND THEIR APPLICATIONS, 99-136 2008

**Alvarez M, Martin PP** Higher-dimensional Temperley-Lieb algebras *Journal of Physics A: Mathematical and Theoretical*, **40**, 2007

DOI:10.1088/1751-8113/40/38/F01

View abstract

**Green RM, Martin PP** Constructing cell data for diagram algebras *Journal of Pure and Applied Algebra*, **209**, 551-569, 2007

DOI:10.1016/j.jpaa.2006.07.016

View abstract

**Martin P, Green RM, Parker A** Towers of recollement and bases for diagram algebras: planar diagrams and a little beyond *Journal of Algebra*, **316**, 392-452, 2007

DOI:10.1016/j.jalgebra.2007.04.013

**Doikou A, Martin P** On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary *Journal of Statistical Mechanics*, **2006**, pp.P06004

DOI:10.1088/1742-5468/2006/06/P06004

**Elgamal A, Martin P** Ramified partition algebras *Mathematische Zeitschrift*, **246**, 473-500, 2004

DOI:10.1007/s00209-003-0581-4

**Martin P, Woodcock D** Generalized blob algebras and alcove geometry *LMS Journal of Computation and Mathematics*, **6**, 249-296, 2003

**Cox AG, Graham JJ, Martin P** The blob algebra in positive characteristic *Journal of Algebra*, **266**, 584-635, 2003

DOI:10.1016/S0021-8693(03)00260-6

**Green RM, Martin PP, Parker AE** A presentation for the symplectic blob algebra *J. Algebra Appl.*

Green RM; Martin PP; Parker AE *On the non-generic representation theory of the symplectic blob algebra*

**Martin P, Mazorchuk V** Fiat categorification of the symmetric inverse semigroup ISn and the semigroup Fn∗ *Semigroup Forum*, 1-18

DOI:10.1007/s00233-017-9866-5

View abstract

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