Murnaghan-Nakayama rule for the cyclotomic Hecke algebra and applications

Abstract

We establish a Murnaghan--Nakayama rule for the irreducible characters of the cyclotomic Hecke algebra $\mathscr H_{m,n}(q,u)$ on Shoji's standard elements. Combined with Shoji's determinacy result, our formula provides a direct combinatorial route to the full irreducible character table of $\mathscr H_{m,n}(q,u)$. Our construction is based on our recent multi-parameter Murnaghan--Nakayama rule for Macdonald polynomials and specializes uniformly to several previously known formulas, including those for the complex reflection group of type $G(m,1,n)$ and the Iwahori--Hecke algebras of types $A$ and $B$. In a dual framework, using the vertex operator realization of Schur functions, we also derive a complementary iterative formula for irreducible characters on upper multipartitions, which may be viewed as a dual Murnaghan--Nakayama rule. As applications, we obtain a Regev-type formula and a Lübeck--Prasad--Adin--Roichman-type formula for cyclotomic Hecke algebras, extending the corresponding formulas for the Iwahori--Hecke algebra of type $A$ and the complex reflection group, respectively. We further introduce the notion of multiple bitrace for cyclotomic Hecke algebras and give a general combinatorial formula for the multiple bitrace. As a specialization, this yields the second orthogonality relation for irreducible characters of the complex reflection group $W_{m,n}$. For practical computation, we also include in an appendix a SageMath implementation of our Murnaghan--Nakayama rule, which computes individual character values and the full character table.