The fundamental invariant of the Hecke algebra Hn(q) characterizes the representations of Hn(q), S n, SUq(N), and SU(N)

Abstract

The irreducible representations (irreps) of the Hecke algebra Hn(q) are shown to be completely characterized by the fundamental invariant of this algebra, Cn. This fundamental invariant is related to the quadratic Casimir operator, 𝒞2, of SUq(N), and reduces to the transposition class-sum, [(2)]n, of Sn when q → 1. The projection operators constructed in terms of Cn for the various irreps of Hn(q) are well behaved in the limit q → 1, even when approaching degenerate eigenvalues of [(2)]n. In the latter case, for which the irreps of Sn are not fully characterized by the corresponding eigenvalue of the transposition class-sum, the limiting form of the projection operator constructed in terms of Cn gives rise to factors that depend on higher class-sums of Sn, which effect the desired characterization. Expanding this limiting form of the projection operator into a linear combination of class-sums of Sn, the coefficients constitute the corresponding row in the character table of Sn. The properties of the fundamental invariant are used to formulate a simple and efficient recursive procedure for the evaluation of the traces of the Hecke algebra. The closely related quadratic Casimir operator of SUq(N) plays a similar role, providing a complete characterization of the irreps of SUq(N) and—by constructing appropriate projection operators and then taking the q → 1 limit—those of SU(N) as well, even when the quadratic Casimir operator of the latter does not suffice to specify its irreps.