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

J. Katriel, B. Abdesselam, A. Chakrabarti
1995 Journal of Mathematical Physics  
The irreducible representations (irreps) of the Hecke algebra H_n(q) are shown to be completely characterized by the fundamental invariant of this algebra, C_n. This fundamental invariant is related to the quadratic Casimir operator, C_2, of SU_q(N), and reduces to the transposition class-sum, [(2)]_n, of S_n when q→ 1. The projection operators constructed in terms of C_n for the various irreps of H_n(q) are well-behaved in the limit q→ 1, even when approaching degenerate eigenvalues of
more » ... In the latter case, for which the irreps of S_n are not fully characterized by the corresponding eigenvalue of the transposition class-sum, the limiting form of the projection operator constructed in terms of C_n gives rise to factors that depend on higher class-sums of S_n, which effect the desired characterization. Expanding this limiting form of the projection operator into a linear combination of class-sums of S_n, the coefficients constitute the corresponding row in the character table of S_n. 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 SU_q(N) plays a similar role, providing a complete characterization of the irreps of SU_q(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.
doi:10.1063/1.531218 fatcat:lyauv3ayuvcvnoco77t5ifwuka