The fundamental invariant of the Hecke algebra $H_n(q)$ characterizes the representations of $H_n(q)$, $S_n$, $SU_q(N)$ and $SU(N)$

J. Katriel; B. Abdesselam; A. Chakrabarti

doi:10.1063/1.531218

paper

The fundamental invariant of the Hecke algebra $H_{n} (q)$ characterizes the representations of $H_{n} (q)$ , $S_{n}$ , $S U_{q} (N)$ and $S U (N)$

arXiv:q-alg/9501021 · doi:10.1063/1.531218

Abstract

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 $S U_{q} (N)$ , and reduces to the transposition class-sum, $[(2)]_{n}$ , of $S_{n}$ when $q \to 1$ . The projection operators constructed in terms of $C_{n}$ for the various irreps of $H_{n} (q)$ are well-behaved in the limit $q \to 1$ , even when approaching degenerate eigenvalues of $[(2)]_{n}$ . 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 $S U_{q} (N)$ plays a similar role, providing a complete characterization of the irreps of $S U_{q} (N)$ and - by constructing appropriate projection operators and then taking the $q \to 1$ limit - those of $S U (N)$ as well, even when the quadratic Casimir operator of the latter does not suffice to specify its irreps.

32 pages, Latex-file, Tables in a Latex form are included at the end of the file

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The fundamental invariant of the Hecke algebra $H_{n} (q)$ characterizes the representations of $H_{n} (q)$ , $S_{n}$ , $S U_{q} (N)$ and $S U (N)$