dorsal/arxiv
View SchemaThe fundamental invariant of the Hecke algebra $H_n(q)$ characterizes the representations of $H_n(q)$, $S_n$, $SU_q(N)$ and $SU(N)$
| Authors | J. Katriel, B. Abdesselam, A. Chakrabarti |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9501021 |
| URL | https://arxiv.org/abs/q-alg/9501021 |
| DOI | 10.1063/1.531218 |
| Journal | J.Math.Phys. 36 (1995) 5139-5158 |
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, ${\cal{C}}_2$, of $SU_q(N)$, and reduces to the transposition class-sum, $[(2)]_n$, of $S_n$ when $q\rightarrow 1$. The projection operators constructed in terms of $C_n$ for the various irreps of $H_n(q)$ are well-behaved in the limit $q\rightarrow 1$, 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 $SU_q(N)$ 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 $q\rightarrow 1$ limit - those of $SU(N)$ as well, even when the quadratic Casimir operator of the latter does not suffice to specify its irreps.
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"abstract": "The irreducible representations (irreps) of the Hecke algebra $H_n(q)$ are\nshown to be completely characterized by the fundamental invariant of this\nalgebra, $C_n$. This fundamental invariant is related to the quadratic Casimir\noperator, ${\\cal{C}}_2$, of $SU_q(N)$, and reduces to the transposition\nclass-sum, $[(2)]_n$, of $S_n$ when $q\\rightarrow 1$. The projection operators\nconstructed in terms of $C_n$ for the various irreps of $H_n(q)$ are\nwell-behaved in the limit $q\\rightarrow 1$, even when approaching degenerate\neigenvalues of $[(2)]_n$. In the latter case, for which the irreps of $S_n$ are\nnot fully characterized by the corresponding eigenvalue of the transposition\nclass-sum, the limiting form of the projection operator constructed in terms of\n$C_n$ gives rise to factors that depend on higher class-sums of $S_n$, which\neffect the desired characterization. Expanding this limiting form of the\nprojection operator into a linear combination of class-sums of $S_n$, the\ncoefficients constitute the corresponding row in the character table of $S_n$.\nThe properties of the fundamental invariant are used to formulate a simple and\nefficient recursive procedure for the evaluation of the traces of the Hecke\nalgebra. The closely related quadratic Casimir operator of $SU_q(N)$ plays a\nsimilar role, providing a complete characterization of the irreps of $SU_q(N)$\nand - by constructing appropriate projection operators and then taking the\n$q\\rightarrow 1$ limit - those of $SU(N)$ as well, even when the quadratic\nCasimir operator of the latter does not suffice to specify its irreps.",
"arxiv_id": "q-alg/9501021",
"authors": [
"J. Katriel",
"B. Abdesselam",
"A. Chakrabarti"
],
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"q-alg",
"math.QA"
],
"doi": "10.1063/1.531218",
"journal_ref": "J.Math.Phys. 36 (1995) 5139-5158",
"title": "The fundamental invariant of the Hecke algebra $H_n(q)$ characterizes the representations of $H_n(q)$, $S_n$, $SU_q(N)$ and $SU(N)$",
"url": "https://arxiv.org/abs/q-alg/9501021"
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