×

Littlewood-Richardson coefficients for Hecke algebras at roots of unity. (English) Zbl 0714.20004

The classical Littlewood-Richardson coefficients are the structure constants of the algebra of symmetric functions in the basis of Schur functions. This algebra is isomorphic to \(\oplus_ nK_ 0(CS_ n)\) with multiplication given by \([p_ 1]*[p_ 2]=[p_ 1shift_ m(p_ 2)]\), where \(p_ 1\), \(p_ 2\) are idempotents in \(CS_ m\), \(CS_ n\) and \(shift_ m\) is the homomorphism \(CS_ n\to CS_{m+n}\) which takes the transposition (i i\(+1)\) to \((i+m i+m+1).\)
A similar algebra can be constructed whenever one has an analogue of the shift operation. In particular, this can be done with the braid group algebra \(CB_{\infty}\) (more precisely, with its approximately finite- dimensional representations) and with its quotients the Hecke algebras H(q). If q is not a root of unity, the structure constants of the resulting algebra turn out to be the same as the classical Littlewood- Richardson coefficients. If q is a primitive l-th root of unity, \(H_ n(q)\) is not semisimple, but one can define an approximately finite- dimensional quotient \(H_ n^{k,l}\) of \(H_ n\) for each \(0\leq k<l\). The structure constants of the algebras associated to these quotients are the coefficients in the title of the paper.
The authors find a formula for these q Littlewood-Richardson coefficients (when q is a root of unity) in terms of the classical ones. It turns out to be the same as the formula for the fusion coefficients in the WZW conformal field theory. It is probably also related to tensor products of representations of the quantum group \(sl(k)_ q\).
Reviewer: A.N.Pressley

MSC:

20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
81T08 Constructive quantum field theory
05A19 Combinatorial identities, bijective combinatorics
20F36 Braid groups; Artin groups
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16S34 Group rings
20C30 Representations of finite symmetric groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Birman, J., Braids, links and mapping class groups, (Ann. Math. Studies No. 82 (1974), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ)
[2] Birman, J.; Wenzl, H., Braids, link polynomials and a new algebra, Transactions Amer. Math. Soc., 313, 249-273 (1989) · Zbl 0684.57004
[3] Bourbaki, N., Goupes et Algèbres de Lie (1982), Masson: Masson Paris, Chaps. IV-I · Zbl 0505.22006
[4] Brauer, R., On algebras which are connected with the semisimple continuous groups, Ann. of Math., 63, 854-872 (1937) · JFM 63.0873.02
[5] Dixmier, J., \(C∗\) Algebras (1977), North-Holland, [French original: Gauthier-Villars, Paris, 1964] · Zbl 0366.17007
[6] Jimbo, M., A \(q\)-analogue of \(u\)(g \(l(N + 1))\), Hecke algebra and the Yang-Baxter equation, Lett. Math. Phys., 11, 247-252 (1986) · Zbl 0602.17005
[7] Jones, V. F.R, Index for subfactors, Invent. Math., 72, 1-25 (1983) · Zbl 0508.46040
[8] Handelman, D., Positive polynomials and product type actions of compact groups, Mem. Amer. Math. Soc., 54, No. 320 (1985) · Zbl 0571.46045
[9] V. Kac; V. Kac · Zbl 0716.17022
[10] Kuniba, A.; Nakanishi, T., Level-Rank duality in fusion RSOS models (1990), Kyushu University, preprint · Zbl 0748.17031
[11] Littlewood, D.; Richardson, A. R., Group characters and algebra, Philos. Trans. Roy. Soc. London Ser. A, 233, 49-141 (1934) · Zbl 0009.20203
[12] MacDonald, I., Symmetric Functions and Hall polynomials (1979), Oxford Univ. Press (Clarendon): Oxford Univ. Press (Clarendon) London/New York · Zbl 0487.20007
[13] Pressley, A.; Segal, G., Loop Groups (1986), Oxford Univ. Press (Clarendon): Oxford Univ. Press (Clarendon) London/New York · Zbl 0618.22011
[14] Remmel, J.; Whitney, R., Multiplying Schur functions, J. Algorithms, 5, 471-487 (1984) · Zbl 0557.20008
[15] Rosso, M., Finite dimensional representations of the quantum analog of the enveloping algebra of a complex semisimple Lie algebra, Comm. Math. Phys., 117, 581-593 (1988) · Zbl 0651.17008
[16] Tsuchiya, A.; Kanie, Y., Vertex operators in conformal field theory on \(P^1\) and monodromy representations of braid groups, Adv. Stud. Pure Math., 16, 297-372 (1988)
[17] Verlinde, E., Fusion rules and modular transformations in 2D conformal field theory, Nuclear Phys. Ser. B, 300, 360-376 (1988) · Zbl 1180.81120
[18] Wassermann, A. J., Automorphic Actions of Compact Groups on Operator Algebras, (Thesis (1980), University of Pennsylvania)
[19] Walton, M. A., Algorithm for the WZW fusion rules: a proof (1990), Université Laval: Université Laval Québec, preprint
[20] Wenzl, H., Hecke algebras of type \(A_n\) and subfactors, Invent. Math., 92, 349-383 (1988) · Zbl 0663.46055
[21] Wenzl, H., On the structure of Brauer’s centralizer algebras, Ann. of Math., 128, 2, 173-193 (1988) · Zbl 0656.20040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.