Murphy operators and the centre of the Iwahori-Hecke algebras of type \(A\)
This paper studies the question of whether the centre of the Iwahori-Hecke algebra \({\mathcal H}_{R,q}({\mathcal S}_n)\) of the symmetric group is the set of symmetric polynomials in the Murphy operators. Dipper and James have conjectured that this is indeed the case for an arbitrary base ring \(R\) containing an arbitrary invertible element \(q\), and they have proved the conjecture for \(\mathcal H\) semisimple although their proof contains a gap for the general case. The current paper reduces the problem to the combinatorial problem of checking an orthogonality property for a certain (previously unknown) set of polynomials. The author also considers a stronger conjecture due to James regarding the determinant of a certain matrix of inner products and offers a conjecture for the inverse of the matrix.
scientific article; zbMATH DE number 1334707