zbMATH — the first resource for mathematics

Euler-Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras. (English) Zbl 0835.05085
The Iwahori-Hecke algebra \(H_n\) associated to the symmetric group \(S_n\) has a faithful representation as an algebra of operators of the polynomial algebra in \(n\) variables such that the symmetric polynomials are treated as scalars under the action of \(H_n\). These symmetrizing operators are the Newton divided differences and their deformations. The simple operators satisfy the Yang-Baxter relations. The authors give several expressions of the operators corresponding to a maximal permutation and recover the generalized Euler-Poincaré characteristic defined by Hirzebruch in the geometry of flat manifolds. Restricting the action of the Hecke algebra to weight spaces, the authors also recover one of the usual descriptions of its representations. They also obtain \(q\)-idempotents and give a \(q\)-analogue of the Specht representations as orbits of products of \(q\)-Vandermonde functions. The authors study different constructions of irreducible representations corresponding to hook partitions and describe them in terms of Kazhdan-Lustig graphs. This interpretation is applied to the diagonalization of the Hamiltonian of a quantum spin chain with the quantum superalgebra \(U_q ({\mathfrak su} (1/1))\) as a symmetry algebra.
Reviewer: V.Drensky (Sofia)

05E10 Combinatorial aspects of representation theory
20C30 Representations of finite symmetric groups
14M15 Grassmannians, Schubert varieties, flag manifolds
20G45 Applications of linear algebraic groups to the sciences
Full Text: DOI
[1] Aitken, A. C., On induced permutation matrices and the symmetric group, Proc. Edinburgh Math. Soc., (2) 5 (1936), 1-13. · Zbl 0015.24803 · doi:10.1017/S0013091500008208
[2] Bernstein, I. N., Gelfand, I. M. and Gelfand, S., Schubert cells and cohomology of the spaces G/P, Russian Math. Surveys, 28 (1973), 1-26. · Zbl 0289.57024 · doi:10.1070/rm1973v028n03ABEH001557
[3] Carre, C., Lascoux, A. and Leclerc, B., Turbo-straigthening for decomposition into standard buses, Internal. J. Alg. Comp., 2 (1992), 275-290. · Zbl 0773.20001 · doi:10.1142/S0218196792000165
[4] Cherednik, I. V., On R-matrix quantization of formal loop groups, in Group theoretical methods in physics, Proc. 3rd Semin., Yurmala (USSR), 1985, 2 (1986), 161-180. · Zbl 0674.17002
[5] , Quantum groups as hidden symmetries of classic representation theory, in Differential geometric methods in theoretical physics (A.I. Solomon, Ed), World Scientific, Singapore, 1989, pp. 47-54.
[6] Deguchi, T. and Akutsu, Y., Graded solutions of the Yang-Baxter relation and link polynomials,/. Phys. A, 23 (1990), 1861-1875. · Zbl 0724.57002 · doi:10.1088/0305-4470/23/11/014
[7] Demazure, M., Desingularisation des varietes de Schubert, Ann. Sci. EC. Norm. Sup., 6 (1974), 53-88. · Zbl 0312.14009 · numdam:ASENS_1974_4_7_1_53_0 · eudml:81930
[8] Dipper, R. and James, G., Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc., 54 (1987), 57-82. · Zbl 0615.20009 · doi:10.1112/plms/s3-54.1.57
[9] Fomin, S. and Kirillov, A. N., The Yang-Baxter equation, symmetric functions and Schubert polynomials, preprint Inst. Mittag-Leffler (1992).
[10] , Yang-Baxter equation, symmetric functions and Grothendieck polynomials, preprint (1993).
[11] Grothendieck, A., Theoreme de Riemann-Roch, in Seminaire de Geometric Algebrique 6, Springer Lecture Notes in Math., 225 (1971).
[12] Gyoja, A., A ^-analogue of Young symmetrizer, Osaka J. Math., 23 (1986), 841-852. · Zbl 0644.20012
[13] Hamermesh, M., Group theory and its applications to physical problems, Addison-Wesley, Reading Mass., 1962. · Zbl 0100.36704
[14] Hirzebruch, F., Topological methods in algebraic geometry, Springer, Berlin, 1966. · Zbl 0138.42001
[15] Hirzebruch, F., Berger, T. and Jung, R., Manifolds and Modular Forms, Vieweg, Wiesbaden, (1992). · Zbl 0767.57014
[16] Hinrichsen, H. and Rittenberg, V., A two parameter deformation of the 5(7(1/1) superalgebra and the XY quantum chain in a magnetic field, preprint CERN-TH. 6299/91, 1991.
[17] James, G. D. and Kerber, A., The Representation Theory of the Symmetric Group, Addison-Wesley, Reading Mass., 1981. · Zbl 0491.20010
[18] Jones, V., Hecke algebra representations of braid groups and link polynomials, Ann. of Math., 126 (1987), 335-388. · Zbl 0631.57005 · doi:10.2307/1971403
[19] King, R. C. and Wybourne, B. G., Representations and traces of the Hecke algebras Hn(q) of type A < -i,/. Math. Phys., 33 (1992), 4-14. · Zbl 0752.05058 · doi:10.1063/1.529925
[20] Lakshmibai, V. and Sheshadri, C. S., Singular locus of a Schubert variety, Bull. Amer. Math. Soc., (II) 2 (1984), 363-366. · Zbl 0549.14016 · doi:10.1090/S0273-0979-1984-15309-6
[21] Lascoux, A., Caracteristique d’Euler-Poincare et produit des caracteres, C. R. Acad. Sci. Paris Ser. I, 299 (1984), 447-450. · Zbl 0577.20009
[22] Lascoux, A. and Schtitzenberger, M. P., Polynomes de Schubert, C. R. Acad. Sci. Paris, 294 (1982), 447-450. · Zbl 0495.14031
[23] , Symmetrization operators on polynomial rings, Fund. Anal Appl, 21 (1987), 77-78. · Zbl 0659.13008 · doi:10.1007/BF01077811
[24] Macdonald, I. G., Symmetric functions and Hall polynomials, Oxford Math. Monographs, Clarendon Press, Oxford, 1979. · Zbl 0487.20007
[25] Martin, P., Potts models and related problems in statistical mechanics, World Scientific, Singapore, 1991. · Zbl 0734.17012
[26] , The structure of n variables polynomial rings as Hecke algebra modules, J. Phys. A, 26 (1993), 7311-7324. · Zbl 0833.20018 · doi:10.1088/0305-4470/26/24/009
[27] Martin, P. and Rittenberg, V., A template for quantum spin chain spectra, Preprint RIMS Kyoto (1991). · Zbl 0925.17004 · doi:10.1142/S0217751X92003999
[28] Ram, A., A Frobenius formula for the characters of Hecke algebras, Invent. Math., 106 (1991), 461-488. · Zbl 0758.05099 · doi:10.1007/BF01243921 · eudml:143950
[29] Rogawski, J. D., On modules over the Hecke algebra of a p-adic group, Invent. Math., 79 (1985), 443-465. · Zbl 0579.20037 · doi:10.1007/BF01388516 · eudml:143205
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.