Indecomposable tilting modules for the blob algebra

Abstract

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 Ã1 .

The nil-blob algebra: An incarnation of type $\tilde{A}_1$ Soergel calculus and of the truncated blob algebra

“…In [22], relation (4.4) is formulated using the condition i 2 = i 1 − 1. On the other hand, as pointed out in Remark 1.4 of [9], this sign change is irrelevant. Indeed, let B n (m) be the algebra defined by the relations of [22].…”

“…Multiplying with the corresponding U i 's on T λ B λ we get the diagram (2.35) Suppose now that we want to produce the blob diagram from (2.7). Then we need a mark on the first through line and thus we multiply below with a diagram of the form (2.32) with i = 8 which gives us (−2) 9 (2.36) settling the third block, at least up to a unit in F. The algorithm now goes on with the second block, etc. The Theorem is proved.…”

“…We now explain an algorithm for producing a reduced expression for the elements d(t). This algorithm has already been used in the previous papers [22], [9], [6] and [15].…”

“…u i and u i ). For example, in the situation (5.3) we have that H 0 = s 2 s 4 s 6 s 3 s 5 s 4 , H 1 = s 9 s 11 s 10 , U 1 = s [8,12] s [7,13] s [6,14] s [5,15] s [6,14] s [7,13] s [8,12] s [9,11] s [10,10] (5.5)…”

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

“…Namely the blob algebra B κ d is a graded cellular algebra with graded cellular basis indexed by standard tableaux of one-column bipartitions of d. The graded decomposition numbers for the blob algebra were computed by Plaza [24]. Moreover the blob algebra is quasi-hereditary and in that setting Hazi, Martin and Parker [13] determined the structure of the indecomposable tilting modules using the graded structure.…”