Murphy bases of generalized Temperley-Lieb algebras

Abstract

In this article we show how to use some results of G. E. Murphy on the socalled standard basis of Hecke-Algebras of Type A to derive a similar basis for generalized Temperley-Lieb algebras. This standard basis is compared to the usual diagrammatic basis of the original Temperley-Lieb algebra used in knot theory and statistical physics.

Annihilators of Permutation Modules

“…This implies the statement in general, since whenever λ ⊲ µ, one gets from µ to λ by a finite succession of such box-raising operations. Now, for λ ⊲ µ and λ, µ adjacent, we have λ = µ + ε i − ε j , and the result follows from Lemmas 6.2 and 6.3. , using a refinement of an argument of [8]. The following result will be crucial in the proof of Theorem 7.3.…”

“…Since x λ and y λ are invariant under * , it follows that (8) x for any pair s, t of row-standard λ-tableaux. For any λ ⊢ n let Tab(λ) be the set of standard λ-tableaux, and set…”

“…Theorem 5.2 is easily derived from a lemma of [8] along with properties of cellular bases. To prove Theorem 7.3 we exploit Schur-Weyl duality between H and the quantized enveloping algebra U(gl n ).…”

Cellular Bases for Algebras with a Jones Basic Construction

“…The cellularity of Jones-Temperley-Lieb algebras was established by Graham and Lehrer [16]. Härterich [18] has given Murphy bases for generalized Temperley-Lieb algebras. Let S be an integral domain and δ ∈ S. The Jones-Temperley-Lieb algebra A n = A n (S; δ) is the unital S-algebra presented by the generators e 1 , .…”

Schur-Weyl duality in positive characteristic

“…The author is grateful to Jun Hu for bringing reference [12] to his attention, and to the referee for useful suggestions. where End Sr (V ⊗r ) (respectively, End Γ (V ⊗r )) is defined to be the algebra of linear operators on V ⊗r commuting with all operators in Φ(S r ) (respectively, Ψ(Γ)).…”