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Polynomial representations of the Hecke algebra of the symmetric group. (English) Zbl 1293.20009

In the article under review, the author uses a generalization of the Yang-Baxter graph for the irreducible representations of the symmetric group to construct a polynomial basis (and an adjoint basis) for each irreducible representation of the Hecke algebra of the symmetric group. This basis satisfies easy vanishing properties, and decompositions can be obtained by specializations. In addition, each polynomial in the basis is a simultaneous eigenfunction of the Jucys-Murphy elements.

MSC:

20C08 Hecke algebras and their representations
05E10 Combinatorial aspects of representation theory
20C30 Representations of finite symmetric groups
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[1] H. Barcelo and A. Ram, New Perspectives in Algebraic Combinatorics, Mathematical Sciences Research Institute Publications 38 (Cambridge University Press, Cambridge, 1999) pp. 23–90.
[2] DOI: 10.1142/S0218196792000165 · Zbl 0773.20001 · doi:10.1142/S0218196792000165
[3] DOI: 10.1017/CBO9781139192361 · doi:10.1017/CBO9781139192361
[4] DOI: 10.1007/BF01077327 · Zbl 0599.20050 · doi:10.1007/BF01077327
[5] DOI: 10.1016/j.jcta.2011.08.002 · Zbl 1232.05232 · doi:10.1016/j.jcta.2011.08.002
[6] DOI: 10.1016/0012-365X(80)90062-X · Zbl 0451.20012 · doi:10.1016/0012-365X(80)90062-X
[7] Di Francesco P., Electron. J. Combin. 12 pp 27– (2005)
[8] Garnir H., Mémoires de la Soc. Royale des Sc. de Liège 10, in: Théorie de la représentation des groupes symétriques (1950)
[9] James G., Encyclopedia of Mathematics and its Applications 16, in: The Representation Theory of the Symmetric Group (1981)
[10] DOI: 10.1007/s00220-007-0341-0 · Zbl 1136.82008 · doi:10.1007/s00220-007-0341-0
[11] DOI: 10.1007/BF01390031 · Zbl 0499.20035 · doi:10.1007/BF01390031
[12] DOI: 10.4171/CMH/72.1.7 · Zbl 0954.05049 · doi:10.4171/CMH/72.1.7
[13] DOI: 10.1007/BF02558465 · Zbl 0928.05061 · doi:10.1007/BF02558465
[14] DOI: 10.1007/s00026-001-8019-3 · Zbl 0988.05093 · doi:10.1007/s00026-001-8019-3
[15] DOI: 10.1090/cbms/099 · doi:10.1090/cbms/099
[16] DOI: 10.1007/s10114-009-6535-y · Zbl 1230.05280 · doi:10.1007/s10114-009-6535-y
[17] DOI: 10.1007/BFb0063238 · doi:10.1007/BFb0063238
[18] DOI: 10.1007/BF01077996 · Zbl 0625.20056 · doi:10.1007/BF01077996
[19] Littlewood D. E., The Theory of Group Characters and Matrix Representations of Groups (1940) · JFM 66.0093.02
[20] DOI: 10.1016/0021-8693(92)90045-N · Zbl 0794.20020 · doi:10.1016/0021-8693(92)90045-N
[21] DOI: 10.1007/BF02433451 · Zbl 0959.20014 · doi:10.1007/BF02433451
[22] Rota G. C., Combinatorial Theory and Invariant Theory (1971) · Zbl 0362.05044
[23] Rutherford D. E., Substitutional Analysis (1948) · Zbl 0038.01602
[24] DOI: 10.1155/S107379289600030X · Zbl 0861.05063 · doi:10.1155/S107379289600030X
[25] DOI: 10.1007/BF01201387 · Zbl 0011.10301 · doi:10.1007/BF01201387
[26] DOI: 10.1215/S0012-7094-41-00852-9 · Zbl 0061.04102 · doi:10.1215/S0012-7094-41-00852-9
[27] A. Vershik, Topics in Algebra, Banach Center Publications 26 (Polish Academy of Sciences, 1990) pp. 467–473.
[28] Weyl H., The Classical Groups, Their Invariants and Their Representations (1946) · Zbl 1024.20502
[29] DOI: 10.1103/PhysRevLett.19.1312 · Zbl 0152.46301 · doi:10.1103/PhysRevLett.19.1312
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