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1996 | 36 | 1 | 61-70
Tytuł artykułu

Intertwining spaces associated with q-analogues of the Young symmetrizers in the Hecke algebra

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Abstrakty
EN
Abstract. Let H be the Hecke algebra of the symmetric group. With each subset S ⊂ [1,n-1], we associate two idempotents $□_S$ and $∇_S$ which are q-deformations of the symmetrizer and antisymmetrizer relative to the Young subgroup ${\goth S}_{I_S}$ generated by the simple transpositions ${(i,i+1)}_{i ∈ S}$. We give here explicit bases for the intertwining space $□_{S_1}H ∇_{S_2}$, indexed by the double classes ${\goth S}_{I_{S_1}}\{\goth S}_n/{\goth S}_{I_{S_2}}$. We also compute bases and characters of the right ideals {\goth I}(I,J)= □_{S_1}H ∇_{S_2}H.
FR
Résumé. Soit H, l'algèbre de Hecke du groupe symétrique. À chaque sous ensemble S ⊂ [1,n-1], on associe deux idempotents $□_S$ et $∇_S$ qui sont les q-déformations des symétriseur et antisymétriseur du sous groupe de Young ${\goth S}_{I_S}$ engendré par les transpositions simples ${(i,i+1)}_{i ∈ S}$. Nous donnons ici des bases explicites pour le sous espace d'entrelacement $□_{S_1}H ∇_{S_2}$, indexées par les doubles classes ${\goth S}_{I_{S_1}}\{\goth S}_n/{\goth S}_{I_{S_2}}$. Nous calculons également des bases et les caractères des idéaux {\goth I}(I,J)= □_{S_1}H ∇_{S_2}H.
Słowa kluczowe
Rocznik
Tom
36
Numer
1
Strony
61-70
Opis fizyczny
Daty
wydano
1996
Twórcy
  • L.I.R., Université de Rouen, Place E. Blondel, B.P. 118, 76134 Mont-Saint-Aignan Cedex, France
autor
  • L.I.T.P., Université Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, France
Bibliografia
  • [1] G. E. Andrews, The theory of partitions, Addison-Wesley, London, 1976.
  • [2] C. W. Curtis, I. Reiner, Representations theory of finite groups and associative algebras, Interscience, New York, 1962.
  • [3] R. Dipper and G. James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 54 (1987), 57-82.
  • [4] G. Duchamp, D. Krob, B. Leclerc, A. Lascoux, T. Scharf, J. Y. Thibon, Euler-Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras, Publ. Res. Inst. Math. Sci. 31 (1995), 179-201.
  • [5] W. Fulton and J. Harris, Representation Theory, a first course, Graduate Texts in Mathematics 129, Springer, New York, 1991.
  • [6] A. Gyoja, A q-analogue of Young symmetrizer, Osaka J. Math. 23 (1986), 841-852.
  • [7] P. N. Hoefsmith, Representations of Hecke algebras of finite groups with BN-pairs of classical type, Ph.D. Thesis, Univ. of British Columbia, 1974.
  • [8] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, Cambridge, 1990.
  • [9] G. James, A. Kerber, The Representation Theory of the Symmetric Group, %Cambridge University Press. Encyclopedia of Mathematics and its Application 16, Addison-Wesley, Reading, 1981.
  • [10] A. A. A. Jucys, Symmetric polynomials and the center of the symmetric group ring, Institute of Physics and Mathematics of the Academy of Sciences, Lithuanian SSR, USSR, Reports on Mathematical Physics 5 (1974).
  • [11] A. Kerber, Algebraic Combinatorics Via Finite Group Actions, Wissenschaftsverlag, Mannheim, 1991.
  • [12] A. Lascoux and M.-P. Schützenberger, Formulaire Raisonné de fonctions Symétriques, LITP, U.E.R. Maths, Paris VII, Oct. 1985.
  • [13] A. Lascoux and M.-P. Schützenberger, Le monoide plaxique, Quaderni de 'La ricerca scientifica', No. 109, ROMA, CNR, 1981.
  • [14] V. F. Molčanov, On matrix elements of irreducible representations of a symmetric group, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 21 (1966), No. 1, 52-57.
  • [15] G. E. Murphy, The idempotents of the symmetric group and Nakayama's conjecture, J. Algebra 81 (1983), 258-265.
  • [16] G. E. Murphy, On the Representation theory of the Symmetric Groups and Associated Hecke Algebras, J. Algebra 152 (1992), 492-513.
  • [17] D. E. Rutherford, Substitutional Analysis, University Press, Edinburgh, 1948.
Typ dokumentu
Bibliografia
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