Potts Models and Related Problems in Statistical Mechanics

The following sections are included:

• Preface

• Acknowledgments

• Contents

The following sections are included:

• On layout and objectives

• Statistical mechanics
  • Partition functions and Hamiltonians
  • Defining lattices
  • Other statistical mechanical functions

• Potts models

• Phase transitions
  • Order parameters
  • Critical exponents

• Dichromatic polynomials

• High and low temperature series

• Block spin renormalisation
  • Fixed points

The following sections are included:

• Partition vectors
  • On internalising a common boundary
  • Local transfer matrices

• Algebraic formulation
  • General local interactions
  • Transfer matrix algebras

• Automorphisms of the lattice

• Perron Frobenius theorem

• The free energy

• Correlation functions

• Spectrum of the transfer matrix

• Potts models

• Standard transfer matrices and duality
  • Alternative layering directions

• Spectrum inversion transformations

The following sections are included:

• Yang-Baxter equations
  • Commuting layer transfer matrices

• Algebraic consequences of the relations
  • The braid point
  • Translations revisited

• Alternative layering directions

• The two dimensional Ising model
  • Useful transfer matrix identities for Q = 2
  • Applying the inversion relation
  • Reparameterised YB equation

The following sections are included:

• The two dimensional Ising model
  • Solution by rotations
  • By translation in the layer
  • By translation in the plane

• On conformal field theory

The following sections are included:

• Algebras
  • On specifying algebras
  • Subalgebras and quotients
  • Linear representation theory

• Physics and the radical

• Induction and restriction

• On the structure of algebras
  • Morita equivalence

• Centraliser algebras

• Bialgebras

• Algebraic overview of following chapters
  • On the braid group
  • Algebra morphisms

The following sections are included:

• Review
  • Technical notes

• Preliminary remarks
  • Two faithful diagrammatic representations

• Generic structure of T_k(q)
  • Combinatorial identities
  • Sequence notation
  • A primitive central idempotent in T_n(q)
  • Translation/reflection notation

• Bases for T_n(q)

• Useful identities
  • The word X(d,c)
  • Longest words, module by module
  • The ABF regular representation

The following sections are included:

• More combinatorics and sequences

• Towards the main theorem
  • Indexing T_n(q) modules

• The main theorem

• Proof of main theorem

The following sections are included:

• Introduction

• The Potts Representation
  • Construction
  • The Potts quotient algebra
  • Reducibility of the Potts representation

• Partition representations
  • The bases
  • Computation of S_n(i)
  • Representation type 1
  • Representation type 2
  • Fixed n pregraph dependence of type 2 bases
  • Generic irreducibility of type 2 representations
  • Quotient relations for type 2 representations

• Physical consequences

• Quotient relations: the Potts representation

The following sections are included:

• Review
  • Technical notes

• On the structure of H^k(q)
  • A primitive central idempotent in H_n(q)
  • Another primitive central idempotent
  • The quotient algebras NH_n(q)
  • The word problem
  • Sequence notation

• On bases for H_n(q)
  • A basis for α-permutation representations
  • On irreducible representations

• The generic structure of H_n(q)

• On the non-generic structure of H_n(q)
  • On the structure of NH_n(q)
  • On T_k(q) modules
  • The structure of 2H_k(q)
  • The structure of 3H_k(q) and higher N

• The defining representation of NH_k(q)

The following sections are included:

• Introduction
  • Notation

• Z_Q symmetric models
  • Transfer matrices
  • Simplicia1 Clifford algebras
  • Canonical representations

• Temperley-Lieb subalgebras
  • Lattice gauge model representations
  • Quotient relations for the gauge representation

The following sections are included:

• Zeros of the partition function

• Semi-infinite systems

• Finite lattice results

• Energy, entropy and combinatorics

The following sections are included:

• Homogeneous 6 vertex model
  • Vertex algebras
  • Deriving a model
  • On solving the model

• Asymmetric 6 vertex models
  • Inhomogeneous 6 vertex models
  • Equivalence with the Potts model

• Homogeneous Z_N vertex models

The following sections are included:

• Introduction
  • Cabling quotients of B_n

• Cabling Temperley-Lieb algebras
  • The local relation
  • Other idempotents

The following sections are included:

• Bibliography

• Index