Tied–boxed algebras

We introduce two new algebras that we call tied–boxed Hecke algebra and tied–boxed Temperley–Lieb algebra. The first one is a subalgebra of the algebra of braids and ties introduced by Aicardi and Juyumaya, and the second one is a tied-version of the well known Temperley–Lieb algebra. We study their representation theory and give cellular bases for them. Furthermore, we explore a strong connection between the tied–boxed Temperley–Lieb algebra and the so-called partition Temperley–Lieb algebra given by Juyumaya. Also, we show that both structures inherit diagrammatic interpretations from a new class of monoids that we call boxed ramified monoids. Additionally, we give presentations for the singular part of the ramified symmetric monoid and for the boxed ramified monoid associated to the Brauer monoid.