Cyclotomic blob algebra and its representation theory

A class of finite dimensional associative algebras, called cyclotomic blob algebras, is introduced by means of diagrammatic generators and relations. They are in fact a generalization of Martin and Saleurʹs blob algebras and have as subalgebras cyclotomic Temperley–Lieb algebras defined recently by Rui and Xi. For precise formalization, an equivalent ring theoretical definition of such algebras is also provided. Based upon the diagrammatic definition, we show that cyclotomic blob algebras are both tabular in the sense of R.M. Green and cellular in the sense of Graham and Lehrer, and we thus obtain a description of all the irreducible representations and a criterion for the quasi-heredity of cyclotomic blob algebras. A branching rule for the cell modules of such algebras is also derived. As a by-product, we prove that cyclotomic Temperley–Lieb algebras are tabular as well, and partially recover their cellularity.