arXiv subjects
Publications and source records attributed to P. P. Martin.
We determine the semisimplicity criterion for even partition algebras over the complex field. Specifically we prove that the even/2-tonal partition algebras $P_n^2(\delta)$ over $\mathbb{C}$ are semisimple for all $n$ if and only if parameter $\delta \not\in \mathbb{N}_0$ .
We study categories whose objects are the braid representations, i.e. strict monoidal functors $F\colon B\rightarrow Mat$ from the braid category $B$ to the category of matrices $Mat$. Braid representations are equivalent to solutions to the (constant) Yang-Baxter equation. A major part of the contribution here is to introduce, compare, and contrast suitable notions of isomorphism of representations. A significant contribution here is an extensive range of key examples and counterexamples. Our approach is mainly motivated by the recent classification of charge conserving Yang-Baxter operators, in which the target $Mat$ is replaced by the subcategory $Match^N$. One objective is to understand from the categorical perspective how the classification was facilitated by this change (with the aim of generalising). Progress is made here by observing that the category of functors $MonFun(B,Mat)$ is itself a monoidal category (Theorem 5.3). In addition, we introduce the notion of sub and quotient objects, proving that an object that is both sub and quotient corresponds to an endomorphism in $MonFun(B,Mat)$ (Theorem 5.18). We also observe that objects with target $Match^N$ always have sub and quotient objects. This monoidal category leads us to consider monoidal subcategories whose objects share a given property, giving rise to a new way to see how group-type and involutive solutions, for example, fit into our framework. Another objective is to understand how universal such a restricted target is. Here we give various properties exposing the implications of different choices of equivalence and describe some relationships among them (Theorem 7.18, Theorem 7.9, Conj. 7.19, Conj. 7.22). A key result of this paper is Th.2.9, which shows that, in a `sufficiently free' setting including our case, every monoidal functor is equivalent to a strict one.
This paper reports some advances in the study of the symplectic blob algebra. We find a presentation for this algebra. We find a minimal poset for this as a quasi-hereditary algebra. We discuss how to reduce the number of parameters defining the algebra from 6 to 4 (or even 3) without loss of representation theoretic generality. We then find some non-semisimple specialisations by calculating Gram determinants for certain cell modules (or standard modules) using the good parametrisation defined. We finish by considering some quotients of specialisations of the symplectic blob algebra which are isomorphic to Temperley--Lieb algebras of type $A$.
We show how the treatment of cellularity in families of algebras arising from diagram calculi, such as Jones' Temperley--Lieb wreaths, variants on Brauer's centralizer algebras, and the contour algebras of Cox et al (of which many algebras are special cases), may be unified using the theory of tabular algebras. This improves an earlier result of the first author (whose hypotheses covered only the Brauer algebra from among these families).
Using $U_q[SU(2)]$ tensor calculus we compute the two-point scalar operators (TPSO), their averages on the ground-state give the two-point correlation functions. The TPSOs are identified as elements of the Temperley-Lieb algebra and a recurrence relation is given for them. We have not tempted to derive the analytic expressions for the correlation functions in the general case but got some partial results. For $q=e^{i π/3}$, all correlation functions are (trivially) zero, for $q=e^{i π/4}$, they are related in the continuum to the correlation functions of left-handed and right-handed Majorana fields in the half plane coupled by the boundary condition. In the case $q=e^{i π/6}$, one gets the correlation functions of Mittag's and Stephen's parafermions for the three-state Potts model. A diagrammatic approach to compute correlation functions is also presented.