Iwahori-Hecke algebras of type A at roots of unity

In this paper, we explore the use of path idempotents for the Hecke algebra of type $A$ at roots of unity. For $q$ a primitive $\ell$-th root of unity we obain a non-unital imbedding of (a quotient of) the group algebra of $S_m$ into (a quotient of) the Hecke algebra $H_n(q)$ for certain $m$ and $n$. From this we recover certain instances of irreducibility criteria of Dipper, James, and Mathas, and we derive estimates on the decomposition numbers for the Hecke algebra at roots of unity. The bounds are easily computed, provide a good geometric picture of the pairs of diagrams $\lambda$, $\mu$ for which the decomposition number $d_{\lambda, \mu}$ is non-zero, and also appers to be a useful adjunct to the exact computation of the decomposition numbers.Comment: **Second** substantial revision of previously submitted manuscript. 38 pages, TeX, with figures in EPS, requires macro BoxedEP

Integrality of Homfly (1,1)-tangle invariants

Given an invariant J(K) of a knot K, the corresponding (1,1)-tangle invariant J'(K)=J(K)/J(U) is defined as the quotient of J(K) by its value J(U) on the unknot U. We prove here that J' is always an integer 2-variable Laurent polynomial when J is the Homfly satellite invariant determined by decorating K with any eigenvector of the meridian map in the Homfly skein of the annulus. Specialisation of the 2-variable polynomials for suitable choices of eigenvector shows that the (1,1)-tangle irreducible quantum sl(N) invariants of K are integer 1-variable Laurent polynomials.Comment: 10 pages, including several interspersed figure