Cell structures on the blob algebra

Abstract

Abstract. We consider the r = 0 case of the conjectures by Bonnafé, Geck, Iancu and Lam on cellular structures on the Hecke algebra of type B. We show that this case induces the natural cell structure on the blob algebra b n by restriction to one-line bipartitions.

Graded cellular basis and Jucys-Murphy elements for generalized blob algebras

“…Suppose that d(s s s) = s i1 · · · s iN−1 s iN is the official reduced expression for d(s s s) so that we have ψ d(s s s) = ψ i1 · · · ψ iN−1 ψ iN . We now have from relations (32), (33), (34) and (35) that y k m s s st t t = y k ψ * d(s s s) e(i λ )ψ d(t t t) = ψ iN y k ψ iN−1 · · · ψ i1 e(i λ )ψ d(t t t) if i = i N , i N + 1 ψ iN y k±1 ψ iN−1 · · · ψ i1 e(i λ ) + δψ iN−1 · · · ψ i1 e(i λ ) if i = i N , i N + 1 (163) where δ = 0, ±1. Using relations (32), (33), (34) and (35) once again, we continue commuting the appearing y k±1 's to the right as far as possible, until they meet e(i λ ).…”

Graded decomposition numbers for the blob algebra

“…The basic strategy to find the graded decomposition numbers for b n (m) is to exploit Theorem 2.4 on this subalgebra. We moreover need the known fact that the (ungraded) decomposition numbers of b n (m) are 0 or 1 (See for instance [17,Theorem 5.5…”

“…In the ungraded setting, the decomposition numbers for b n were determined in [13] and [17] using completely different methods. Our approach is essentially combinatorial, and therefore different from those used in the ungraded case.…”

Bases and BGG resolutions of simple modules of Temperley-Lieb algebras of type B

“…The blob algebra is a quotient of the Hecke algebra of type B, hence it controls portion of the representation theory of the affine Hecke algebras. The decomposition numbers for the ungraded blob algebra were determined by Martin and Woodcock [22] and Ryom-Hansen [26]. However the fact that the blob algebra is graded caused a blast in its study.…”