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15
Canonical bases for Hecke algebra quotients
 TO APPEAR IN MATHEMATICAL RESEARCH LETTERS
, 1999
"... We establish the existence of an IC basis for the generalized Temperley–Lieb algebra associated to a Coxeter system of arbitrary type. We determine this basis explicitly in the case where the Coxeter system is simply laced and the algebra is finite dimensional. ..."
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We establish the existence of an IC basis for the generalized Temperley–Lieb algebra associated to a Coxeter system of arbitrary type. We determine this basis explicitly in the case where the Coxeter system is simply laced and the algebra is finite dimensional.
Cellular Algebras Arising from Hecke Algebras of Type H_n
 Math. Zeit
"... We study a finitedimensional quotient of the Hecke algebra of type Hn for general n, using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity of the quotient algebra. We also discuss th ..."
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Cited by 4 (4 self)
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We study a finitedimensional quotient of the Hecke algebra of type Hn for general n, using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity of the quotient algebra. We also discuss the cellular structure of the algebra, with certain restrictions on the ground ring.
Towers of recollement and bases for diagram algebras: planar diagrams and a little beyond
, 2006
"... The recollement approach to the representation theory of sequences of algebras is extended to pass basis information directly through the globalisation functor. The method is hence adapted to treat sequences that are not necessarily towers by inclusion, such as symplectic blob algebras (diagram alge ..."
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Cited by 4 (0 self)
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The recollement approach to the representation theory of sequences of algebras is extended to pass basis information directly through the globalisation functor. The method is hence adapted to treat sequences that are not necessarily towers by inclusion, such as symplectic blob algebras (diagram algebra quotients of the type ˜C Hecke algebras). By carefully reviewing the diagram algebra construction, we find a new set of functors interrelating module categories of ordinary blob algebras (diagram algebra quotients of the typeB Hecke algebras) at different values of the algebra parameters. We show that these functors generalise to determine the structure of symplectic blob algebras, and hence of certain twoboundary TemperleyLieb algebras arising in Statistical Mechanics. We identify the diagram basis with a cellular basis for each symplectic blob algebra, and prove that these algebras are quasihereditary over a field for almost all parameter choices, and generically semisimple. (That is, we give
Tangle and Brauer diagram algebras of type Dn
, 2007
"... Abstract. A generalization of the Kauffman tangle algebra is given for Coxeter type Dn. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the BirmanMurakamiWenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical ca ..."
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Cited by 3 (3 self)
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Abstract. A generalization of the Kauffman tangle algebra is given for Coxeter type Dn. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the BirmanMurakamiWenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which, in our setup, occurs when the Coxeter type is An−1. The proof involves a diagrammatic version of the Brauer algebra of type Dn of which the generalized TemperleyLieb algebra of type Dn is a subalgebra.
Fully commutative Kazhdan–Lusztig cells
, 2001
"... We investigate the compatibility of the set of fully commutative elements of a Coxeter group with the various types of Kazhdan–Lusztig cells using a canonical basis for a generalized version of the Temperley–Lieb algebra. ..."
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Cited by 3 (1 self)
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We investigate the compatibility of the set of fully commutative elements of a Coxeter group with the various types of Kazhdan–Lusztig cells using a canonical basis for a generalized version of the Temperley–Lieb algebra.
The BMW Algebras of Type Dn
, 2007
"... Abstract. The BirmanMurakamiWenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free over Z[δ ±1, l ±1]/(m(1 − δ) − (l − l −1)) of dimension (2 n +1)n!! −(2 n−1 +1)n!, where n!! = 1 ·3···(2n −1). The Brauer algebra of type Dn is a homomorphic ring image and is also semisimple an ..."
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Cited by 2 (2 self)
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Abstract. The BirmanMurakamiWenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free over Z[δ ±1, l ±1]/(m(1 − δ) − (l − l −1)) of dimension (2 n +1)n!! −(2 n−1 +1)n!, where n!! = 1 ·3···(2n −1). The Brauer algebra of type Dn is a homomorphic ring image and is also semisimple and free of the same dimension, but over the ring Z[δ ±1]. A rewrite system for the Brauer algebra is used in upper bounding the dimension of the BMW algebra. As a consequence of our results, the generalized TemperleyLieb algebra of type Dn turns out to be a subalgebra of the BMW algebra of the same type.
A Projection Property for KazhdanLusztig Bases
 Internat. Math. Res. Notices
, 2000
"... . We compare the canonical basis for a generalized Temperley{Lieb algebra of type A or B with the Kazhdan{Lusztig basis for the corresponding Hecke algebra. Introduction Generalized Temperley{Lieb algebras arise as certain quotients of Hecke algebras associated to Coxeter systems in the same way th ..."
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. We compare the canonical basis for a generalized Temperley{Lieb algebra of type A or B with the Kazhdan{Lusztig basis for the corresponding Hecke algebra. Introduction Generalized Temperley{Lieb algebras arise as certain quotients of Hecke algebras associated to Coxeter systems in the same way that the ordinary Temperley{Lieb algebra can be realised as a quotient of the Hecke algebra of type A (see [9]). The nitedimensional generalized Temperley{Lieb algebras were classied by J. Graham [5] into seven innite families: types A, B, D, E, F , H and I. The rst author was supported in part by an award from the Nueld Foundation. Typeset by A M ST E X 1 2 R.M. GREEN AND J. LOSONCZY In [7], we showed that a generalized Temperley{Lieb algebra arising from a Coxeter system of arbitrary type admits a canonical (more precisely, an IC) basis. Such a basis is by denition unique; furthermore, it is analogous, in a manner which can be made precise, to G. Lusztig's canonical basis for the ...
The KazhdanLusztig Basis and the TemperleyLieb Quotient in Type D
 D, J. Algebra
, 2000
"... : Let H be a Hecke algebra associated with a Coxeter system of type D, and let TL be the corresponding Temperley{Lieb quotient. The algebra TL admits a canonical basis, which facilitates the construction of irreducible representations. In this paper, we explain the relationship between the canonical ..."
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: Let H be a Hecke algebra associated with a Coxeter system of type D, and let TL be the corresponding Temperley{Lieb quotient. The algebra TL admits a canonical basis, which facilitates the construction of irreducible representations. In this paper, we explain the relationship between the canonical basis of TL and the Kazhdan{Lusztig basis of H. Key Words: canonical basis; Coxeter group; Hecke algebra; Kazhdan{Lusztig basis; Temperley{Lieb algebra. 2 1. Introduction Let X be a Coxeter graph and let W (X) be an associated Coxeter group with Coxeter generators S(X) and length function `. Let H(X) be the corresponding Hecke algebra. This is an associative, unital algebra over the ring A = Z[v; v 1 ] of Laurent polynomials. The Hecke algebra H(X) has generators T s , one for each s 2 S(X), which are subject to the following relations: T 2 s = (q 1)T s + q, where q = v 2 ; (T s T s 0 ) m = (T s 0 T s ) m if ss 0 has order 2m; and (T s T s 0 ) m T s = (T s 0 T s ) m...
ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En
, 704
"... Abstract. We show that there is a unique Markov trace on the tower of Temperley– Lieb type quotients of Hecke algebras of Coxeter type En (for all n ≥ 6). We explain in detail how this trace may be computed easily using tom Dieck’s calculus of diagrams. As applications, we show how to use the trace ..."
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Abstract. We show that there is a unique Markov trace on the tower of Temperley– Lieb type quotients of Hecke algebras of Coxeter type En (for all n ≥ 6). We explain in detail how this trace may be computed easily using tom Dieck’s calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan– Lusztig polynomials. 1.