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We study the algebras, $J_{l,n}(\delta)$, introduced by Kadar-Martin-Yu in arXiv:1401.1774. We show that these algebras are iterated inflation algebras. We give a set of generators for $J_{l,n}(\delta)$. We show that this algebra satisifies the CMPX axiomatic framework in arXiv:math/0411395 when the base field is the complex numbers and $\delta \ne 0$. We show that $J_{l,n}(\delta)$ is semisimple over the complex numbers if $\delta$ is complex and not real., Comment: 21 pages

The blob algebra is a finite-dimensional quotient of the Hecke algebra of type $B$ which is almost always quasi-hereditary. We construct the indecomposable tilting modules for the blob algebra over a field of characteristic $0$ in the doubly critical case. Every indecomposable tilting module of maximal highest weight is either a projective module or an extension of a simple module by a projective module. Moreover, every indecomposable tilting module is a submodule of an indecomposable tilting module of maximal highest weight. We conclude that the graded Weyl multiplicities of the indecomposable tilting modules in this case are given by inverse Kazhdan-Lusztig polynomials of type $\tilde{A}_1$., Comment: 26 pages, several figures, best viewed in colour

The symplectic blob algebra $b_n$ ($n \in \mathbb{N}$) is a finite dimensional algebra defined by a multiplication rule on a basis of certain diagrams. The rank $r(n)$ of $b_n$ is not known in general, but $r(n)/n$ grows unboundedly with $n$. For each $b_n$ we define an algebra by presentation, such that the number of generators and relations grows linearly with $n$. We prove that these algebras are isomorphic., Comment: final version submitted before publication. arXiv admin note: substantial text overlap with arXiv:0807.4101

This paper reports key advances in the study of the representation theory of the symplectic blob algebra. For suitable specialisations of the parameters we construct four large families of homomorphisms between cell modules. We hence find a large family of non-semisimple specialisations. We find a minimal poset (i.e. least number of relations) for the symplectic blob as a quasi-hereditary algebra., Comment: 45 pages. arXiv admin note: text overlap with arXiv:0807.4101

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., Comment: 41 pages

We provide a method for constructing central idempotents in the Brauer algebra relating to the splitting of certain short exact sequences. We also determine some of the primitive central idempotents, and relate properties of the idempotents to known facts about the representation theory of the algebra., Comment: 19 pages

In this note, we consider the Lyndon--Hochschild--Serre spectral sequence corresponding to the first Frobenius kernel of an algebraic group G, computing the extensions between simple $G$-modules. We state and discuss a conjecture that $E_2=E_\infty$ and provide general conditions for low-dimensional terms on the $E_2$-page to be the same as the corresponding terms on the $E_{\infty}$-page, i.e. its abutment., Comment: 6 pages