Diagram algebras, Hecke algebras and decomposition numbers at roots of unity

Abstract

We prove that the cell modules of the affine Temperley-Lieb algebra have the same composition factors, when regarded as modules for the affine Hecke algebra of type A, as certain standard modules which are defined homologically. En route, we relate these to the cell modules of the Temperley-Lieb algebra of type B, which provides a connection between Temperley-Lieb algebras on n and n − 1 strings. Applications include the explicit determination of some decomposition numbers of standard modules at roots of unity, which in turn has implications for certain Kazhdan-Lusztig polynomials associated with nilpotent orbit closures. The methods involve the study of the relationships among several algebras defined by concatenation of braid-like diagrams and between these and Hecke algebras. Connections are made with earlier work of Bernstein-Zelevinsky on the "generic case" and of Jones on link invariants.  2003 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Nous démontrons que les "modules cellulaires" de l'algèbre de Temperley-Lieb affine ont, regardés comme modules pour l'algèbre de Hecke affine de type A, les mêmes facteurs de composition que certains modules "standards" qui sont définis homologiquement. Au passage, nous relions ces modules aux modules cellulaires pour l'algèbre de Temperley-Lieb de type B. Parmi les applications est la détermination explicite des nombres de décomposition de certains modules standards aux racines de l'unité, qui implique à son tour la détermination de certains polynômes de Kazhdan-Lusztig associés aux clôtures d'orbites nilpotentes. Nos méthodes consistent à étudier les rapports entre certaines algèbres de concaténation de diagrammes de tresses ou analogues, et entre ces algèbres et les algèbres de Hecke. Le travail est aussi relié aux travaux précédents de Bernstein-Zelevinski dans le cas "génerique" et de Jones sur les invariants des entrelacs.  2003 Éditions scientifiques et médicales Elsevier SAS (2.3) LEMMA.-Let W a ⊃ W a be as described above. The map M a /W a → M a / W a is an unramified covering with covering group Z. The proof is easy. Note that the quotient W a /W a is generated by the element τ = c n , (1, 0, 0,. . ., 0) ∈ Sym n Z n , where c n is the n-cycle (12. .. n) ∈ Sym n. This element has the property that τ n lies in the centre Z(W a). It follows from the lemma that M a / W a may be thought of as the quotient of M a /W a by the cyclic group τ .

Commuting Families in Hecke and Temperley-Lieb Algebras

“…(9.6)]) that the representations of the affine Hecke algebra which appear as affine Temperley-Lieb algebra representations are those representations that correspond, under the Deligne-Langlands correspondence [KL], to unipotent elements in GL n with at most two Jordan blocks. The commuting family of elements we use in this paper does not differ significantly from those used in [GL1]. Our results provide an expansion of these elements in terms of the basis of noncrossing diagrams and place the representations studied in [GL1] into a Schur-Weyl duality context.…”

“…The commuting family of elements we use in this paper does not differ significantly from those used in [GL1]. Our results provide an expansion of these elements in terms of the basis of noncrossing diagrams and place the representations studied in [GL1] into a Schur-Weyl duality context. …”

Markov traces on cyclotomic Temperley–Lieb algebras

“…Using (3.2), (2.2) and the recurrence formula for generalized Tchebychev polynomials, we have (4). In order to prove (3), we need to consider three cases as follows.…”

Generalized Blob Algebras and Alcove Geometry

“…In fact, all the finite irreducible representations of these infinite-dimensional algebras may be constructed using [55]. (Although completeness is not shown there, see [28,57].) From the point of view of lattice statistical mechanics, b n also renders the 'seam' boundary conditions (as in [5]; see also [47], for example) of the ice-type model [4] into the algebraic formalism of Temperley and Lieb [68].…”