0 Ju n 19 98 Weights of Markov Traces on Hecke Algebras ∗

We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of type Bn and type Dn. In order to prove the weight formula, we define representations of the Hecke algebra of type B onto a reduced Hecke Algebra of type A. To compute the weights for type D we use the inclusion of the Hecke algebra of type D into the Hecke algebra of type B. Introduction The Hecke algebra of type Bn, Hn(q, Q), is semisimple whenever Q 6= −q , k ∈ {0,±1, . . .± (n − 1)}, and q is not a root of unity. The simple components are indexed by pairs of Young diagrams. Alternatively, these Hecke algebras can be defined as a finite dimensional quotient of the group algebra of the braid group of type B. Motivated by their study of link invariants related to the braid group of type B, Geck and Lambropolou [GL] have defined certain linear traces on the Hecke algebra of type B called Markov traces. Their definition is given inductively. In this paper we give an alternative way of computing this trace. Since the Hecke algebra of type B is semisimple, any linear trace can be written as a weighted linear combination of the irreducible characters (the usual trace). The coefficients in this linear expression are called weights. The weights are equal to the values of the trace at the minimal idempotents. Since the characters are known, it follows that the weights completely determine the trace. The weights are also indexed by pairs of Young diagrams. We have found the weight formula for the Markov trace defined by Geck and Lambropolou [GL] for the Hecke algebra of type B. The weight formula can be written as a product of Schur functions. To prove this formula we construct a homomorphism from the specialization of the Hecke algebra of type B, Hn(q,−q 1), onto a reduced Hecke algebra of type A. Using this homomorphism we obtain that the Markov trace of the Hecke algebra of type B appears as a pullback of the Markov trace of the reduced Hecke algebra of type A. ∗1991 Subject Classification: 05E99