We construct a faithful tensor representation for the Yokonuma-Hecke algebra Y r,n , and use it to give a concrete isomorphism between Y r,n and Shoji's modified Ariki-Koike algebra. We give a cellular basis for Y r,n and show that the Jucys-Murphy elements for Y r,n are JM-elements in the abstract sense. Finally, we construct a cellular basis for the Aicardi-Juyumaya algebra of braids and ties.

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In 2003, Martin and Woodcock noticed a connection between the representation theory of the blob algebra and the Kazhdan-Lusztig polynomials associated with the infinite dihedral group. However, no conceptual explanation for this coincidence has yet been provided. In this study, a possible explanation of this phenomenon is suggested by enunciating a conjecture that relates the endomorphism algebra of Bott-Samelson bimodules to certain subalgebras of the blob algebra obtained by idempotent truncation. Evidence supporting this conjecture is provided.