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Paul P. Martin

Publications and source records attributed to Paul P. Martin.

13 recordsLinked to original sources
arXiv2026

Paravortices: loop braid representations with both generators involutive

We first motivate the study of a certain quotient of the loop braid category, both for the mathematics underpinning recent approaches to topological quantum computation; and as a key example in non-semisimple higher representation theory. For reasons that will become clear, we call this quotient the mixed doubles category, $MD$. Then our main result is a theorem classifying all mixed doubles representations in rank-2. Each representation yields a mixed doubles group representation for every loop braid group $LB_n$, and we are able to analyse the unified linear representation theory of many of these sequences of representations, using a mixture of very classical, classical, and new techniques. In particular this is a motivating example for the `glue' generalisation of charge-conserving representation theory (a form of rigid higher non-semisimplicity) introduced recently.

math.QA↗
arXiv2025

On semisimplicity criteria and non-semisimple representation theory for the Kadar-Yu algebras

The Kadar--Yu algebras are a physically motivated sequence of towers of algebras interpolating between the Brauer algebras and Temperley--Lieb algebras. The complex representation theory of the Brauer and Temperley--Lieb algebras is now fairly well understood, with each connecting in a different way to Kazhdan--Lusztig theory. The semisimple representation theory of the KY algebras is also understood, and thus interpolates, for example, between the double-factorial and Catalan combinatorial realms. However the non-semisimple representation theory has remained largely open, being harder overall than the (already challenging) Brauer case. In this paper we determine generalised Chebyshev-like forms for the determinants of gram matrices of contravariant forms for standard modules. This generalises the root-of-unity paradigm for Temperley--Lieb algebras (and many related algebras); interpolating in various ways between this and the `integral paradigm' for Brauer algebras. The standard module gram determinants give a huge amount of information about morphisms between standard modules, making thorough use of the powerful homological machinery of towers of recollement (ToR), with appropriate gram determinants providing the ToR `bootstrap'. As for the Brauer and TL cases the representation theory has a strongly alcove-geometric flavour, but the KY cases guide an intriguing generalisation of the overall geometric framework.

math.RT↗
arXiv2024

Boltzmann Distribution on "Short" Integer Partitions with Power Parts: Limit Laws and Sampling

The paper is concerned with the asymptotic analysis of a family of Boltzmann (multiplicative) distributions over the set $\check{\varLambda}^{q}$ of strict integer partitions (i.e., with unequal parts) into perfect $q$-th powers. A combinatorial link is provided via a suitable conditioning by fixing the partition weight (the sum of parts) and length (the number of parts), leading to uniform distribution on the corresponding subspaces of partitions. The Boltzmann measure is calibrated through the hyper-parameters $\langle N\rangle$ and $\langle M\rangle$ controlling the expected weight and length, respectively. We study ``short'' partitions, where the parameter $\langle M\rangle$ is either fixed or grows slower than for typical plain (unconstrained) partitions. For this model, we obtain a variety of limit theorems including the asymptotics of the cumulative cardinality in the case of fixed $\langle M\rangle$ and a limit shape result in the case of slow growth of $\langle M\rangle$. In both cases, we also characterize the joint distribution of the weight and length, as well as the growth of the smallest and largest parts. Using these results we construct suitable sampling algorithms and analyse their performance.

math.PR↗
arXiv2024

Decomposition matrices and blocks for the symplectic blob algebra over the complex field

The symplectic blob algebra is a physically motivated quotient of the Hecke algebra $H(\tilde{C}_n)$ with a diagram calculus. We find the blocks for the symplectic blob algebra for all specialisations of its parameters over the complex numbers. We determine Gram determinants for the cell modules with respect to a canonical contravariant form. We show in particular that the algebra is semisimple over the complex numbers unless at least one of the "quantisation" parameters, or the sum or difference of two of these parameters is integral, or the bulk parameter $q$ is a root of unity. We find decomposition numbers in many of the $q$-generic cases.

math.RT↗
arXiv2020

Geometric partition categories: On short Brauer algebras and their blob subalgebras

The main result here gives an algebra(/linear category) isomorphism between a geometrically defined subcategory $J^1_0$ of a short Brauer category $J_0$ and a certain one-parameter specialisation of the blob category $b$. That is, we prove the Conjecture in Remark 6.7 of [14]. We also define a sequence of generalisations $J^i_{i-1}$ of the category $J^1_0$. The connection of $J_0$ with the blob category inspires a search for connections also with its beautiful representation theory. Here we obtain formulae determining the non-semisimplicity condition (generalising the classical `root-of-unity' condition).

math.RT↗
arXiv2017

A generalised Euler-Poincaré formula for associahedra

We derive a formula for the number of flip-equivalence classes of tilings of an $n$-gon by collections of tiles of shape dictated by an integer partition $λ$. The proof uses the Euler-Poincaré formula; and the formula itself generalises the Euler-Poincaré formula for associahedra.

math.CO↗
arXiv2017

The fibres of the Scott map on polygon tilings are the flip equivalence classes

We associate strand diagrams to tilings of surfaces with marked points, generalising Scott's method for triangulations of polygons. We thus obtain a map from tilings of surfaces to permutations of the marked points on boundary components, the {\em Scott map}. In the disk case (polygon tilings) we prove that the fibres of the Scott map are the flip equivalence classes. The result allows us to consider the size of the image as a generalisation of a classical combinatorial problem, and hence to determine the size in low ranks.

math.CO↗
arXiv2014

On geometrically defined extensions of the Temperley-Lieb category in the Brauer category

We define an infinite chain of subcategories of the partition category by introducing the left-height ($l$) of a partition. For the Brauer case, the chain starts with the Temperley-Lieb ($l=-1$) and ends with the Brauer ($l=\infty$) category. The End sets are algebras, i.e., an infinite tower thereof for each $l$, whose representation theory is studied in the paper.

math.RT↗
arXiv2012

On Brauer algebra simple modules over the complex field

This paper gives two results on the simple modules for the Brauer algebra over the complex field. First we describe the module structure of the restriction of all simple modules. Second we give a new geometrical interpretation of Ram and Wenzl's construction of bases for the 'δ-permissible' simple modules.

math.RT↗
arXiv2007

A Temperley-Lieb category for 2-manifolds

Guided by consideration of problems in 2 and 3 dimensional lattice model computation, we are led to define a number of new categories, and functors between these categories and the partition category, culminating in the introduction of two categories generalising the Temperley-Lieb category. We show how to compute practically in these categories, by giving a combinatorial realisation of their (topological) construction.

math-ph↗
arXiv2005

On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary

Motivated by a study of the crossing symmetry of the `gemini' representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to the Yang--Baxter equation satisfying the crossing condition (that is, integrable quantum spin chains). We show that every crossing representation of the Temperley--Lieb algebra appears in this construction, and in particular that this construction builds new representations. We extend these to new representations of the blob algebra, which build new solutions to the Boundary Yang--Baxter equation (i.e. open spin chains with integrable boundary conditions). We prove that the open spin chain Hamiltonian derived from Sklyanin's commuting transfer matrix using such a solution can always be expressed as the representation of an element of the blob algebra, and determine this element. We determine the representation theory (irreducible content) of the new representations and hence show that all such Hamiltonians have the same spectrum up to multiplicity, for any given value of the algebraic boundary parameter. (A corollary is that our models have the same spectrum as the open XXZ chain with nondiagonal boundary -- despite differing from this model in having reference states.) Using this multiplicity data, and other ideas, we investigate the underlying quantum group symmetry of the new Hamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.

hep-th↗
arXiv2003

The Bubble Algebra: Structure of a Two-Colour Temperley-Lieb Algebra

We define new diagram algebras providing a sequence of multiparameter generalisations of the Temperley-Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional Statistical Mechanics. These algebras give a rigorous foundation to the various "multi-colour algebras" of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in Statistical Mechanics. We demonstrate how these algebras may be used to solve the Yang-Baxter equations.

math-ph↗