Idempotents of the Hecke algebra become Schur functions in the skein of the annulus

Abstract

The Hecke algebra H_n contains well known idempotents E_{\lambda} which are indexed by Young diagrams with n cells. They were originally described by Gyoja. A skein theoretical description of E_{\lambda} was given by Aiston and Morton. The closure of E_{\lambda} becomes an element Q_{\lambda} of the skein of the annulus. In this skein, they are known to obey the same multiplication rule as the symmetric Schur functions s_{\lambda}. But previous proofs of this fact used results about quantum groups which were far beyond the scope of skein theory. Our elementary proof uses only skein theory and basic algebra.