. La-taille-d, une partition ? est le nombre de cases du diagramme et est notée |?|. Par définition, la taille d'un m-uplet ? (m) = (? 1 , . . . , ? m ) est |? (m) | :=

. Avec-cette-définition, une m-partition ? (m) est un ensemble de m-cases tel que, pour tout p entre 1 et m, le sous-ensemble consistant en les m-cases ? (m) avec pos(? (m) ) = p forme une partition usuelle

. Soit-n-la-taille-de-la, Nous plaçons maintenant les nombres 1, . . . , n dans les mcases de ? (m) de telle manière que, dans tout diagramme, les nombres dans les m-cases soient en ordre croissant le long des lignes vers la droite

.. Sous-algèbres-alternées-des-algèbres-de-hecke, .. De-hecke, and .. La-bourbaki, 166 IV.4.1 Définition de la sous-algèbre alternée de l'algèbre, p.168

I. Appendice, A Coefficients dans les relations définissantes des algèbres H + (G), p.170

B. Algorithme-de-coxeter and D. , Todd et formes normales pour les sous-groupes alternés de type A, p.178

D. Type, 184 IV.1 Introduction Soient (G, S) un système de Coxeter et G + le sous-groupe alterné de G. Une présentation de G + par générateurs et relations est donnée

B. Etendons-la-signature-au-groupe, 1} tel que ?(g i ) = ?1 pour i = 0, . . . , n ? 1. Son noyau B(G) + := ker(?) est appelé le sous-groupe alterné du groupe B(G) Le groupe B + (G) est engendré par les éléments g i g j , i

. Remarque, Soit ? la surjection naturelle de l'algèbre de groupe de B(G) vers l'algèbre de Hecke H(G) (qui est le quotient de l'algèbre de groupe de B(G) par la relation (IV.4.1)) Rappelons la Z 2 -graduation de H(G) définie par l'involution ?. Nous soulignons le fait que ?? = ??, l'image de l'algèbre de groupe de B + (G) par ? n'appartient pas à H + (G) ; en d'autres mots, la graduation de H(G) n'est pas induite

T. Arakawa and T. Suzuki, Duality between sln(C) and the Degenerate Affine Hecke Algebra, Journal of Algebra, vol.209, issue.1, pp.288-304, 1998.
DOI : 10.1006/jabr.1998.7530

S. Ariki, On the Semi-simplicity of the Hecke Algebra of (Z/rZ) ??? Sn, Journal of Algebra, vol.169, issue.1, pp.216-225, 1994.
DOI : 10.1006/jabr.1994.1280

S. Ariki, N. Jacon, and C. Lecouvey, The modular branching rule for affine Hecke algebras of type A, preprint ArXiv : 1003, p.4838, 2010.

S. Ariki and K. Koike, A Hecke Algebra of (Z/rZ)Sn and Construction of Its Irreducible Representations, Advances in Mathematics, vol.106, issue.2, pp.216-243, 1994.
DOI : 10.1006/aima.1994.1057

R. Baxter, Exactly Solved Models in Statistical Mechanics, 1982.
DOI : 10.1142/9789814415255_0002

I. Bernstein and A. Zelevinsky, Induced representations of reductive ${\germ p}$-adic groups. I, Annales scientifiques de l'??cole normale sup??rieure, vol.10, issue.4, pp.441-472, 1977.
DOI : 10.24033/asens.1333

K. Bremke and G. Malle, Reduced words and a length function for G(e, 1, n), Indag, Mathem, vol.8, pp.453-469, 1997.

F. Brenti, V. Reiner, and Y. Roichman, Alternating subgroups of Coxeter groups, Journal of Combinatorial Theory, Series A, vol.115, issue.5, pp.845-8770702177, 2008.
DOI : 10.1016/j.jcta.2007.10.004

M. Broué, G. Malle, and R. Rouquier, Complex reflection groups, braid groups, Hecke algebras, J. Reine und Angew. Math, vol.500, pp.127-190, 1998.

B. Casselman, The construction of Hecke algebras associated to a Coxeter group, pp.91-102, 2011.
DOI : 10.1090/conm/543/10731

V. Chari and A. Pressley, A guide to quantum groups, 1995.

I. Cherednik, Factorizing particles on a half-line and root systems, english translation in : Theor, Math. Phys, pp.61-977, 1984.

I. Cherednik, On special bases of irreducible finite-dimensional representations of the degenerate affine Hecke algebra, Funct, Analysis Appl, vol.20, pp.87-89, 1986.

I. Cherednik, A new interpretation of Gelfand-Tzetlin bases, Duke Math, J, vol.54, pp.563-577, 1987.

I. Cherednik, Double affine Hecke algebras, London Mathematical Society Lecture Note Series, vol.319, 2005.
DOI : 10.1017/CBO9780511546501

M. Chlouveraki and N. Jacon, Schur elements for the Ariki???Koike algebra and applications, Journal of Algebraic Combinatorics, vol.281, issue.7, pp.291-311, 2012.
DOI : 10.1007/s10801-011-0314-4
URL : https://hal.archives-ouvertes.fr/hal-00596795

H. Coxeter, Discrete Groups Generated by Reflections, The Annals of Mathematics, vol.35, issue.3, pp.588-621, 1934.
DOI : 10.2307/1968753

H. Coxeter, The complete enumeration of finite groups of the form r 2 i = (r i r j ) k ij = 1, J. London Math. Soc, vol.10, pp.21-25, 1935.

H. Coxeter and W. Moser, Generators and relations for discrete groups, 1980.

H. Coxeter and J. Todd, A practical method for enumerating cosets of a finite abstract group, Proc. Edimburgh Math. Soc, vol.5, pp.26-34, 1936.

J. A. De-azcarraga, P. Kulish, and F. Rodenas, Reflection equations andq-Minkowski space algebras, Letters in Mathematical Physics, vol.34, issue.3, pp.173-182, 1994.
DOI : 10.1007/BF00750660

V. Drinfeld, Quantum groups, Proceedings of the International Congress of Mathematicians, pp.798-820, 1986.

N. Enomoto and M. Kashiwara, Symmetric crystals and affine Hecke algebras of type B ArXiv : math, Proc. Japan Acad. 82, pp.131-1360608079, 2006.

K. Eriksson, A Combinatorial Proof of the Existence of the Generic Hecke Algebra and $R$-Polynomials., MATHEMATICA SCANDINAVICA, vol.75, pp.169-177, 1994.
DOI : 10.7146/math.scand.a-12512

P. Etingof and E. Rains, New Deformations of Group Algebras of Coxeter Groups, II, Geometric and Functional Analysis, vol.17, issue.6, pp.1851-18710604519, 2008.
DOI : 10.1007/s00039-007-0642-7

L. Faddeev, N. Reshetikhin, and L. Takhtadzhyan, Quantization of Lie groups and Lie algebras, Algebra i Analiz, pp.178-206, 1989.

C. Gomez, M. Ruiz-altaba, and G. Sierra, Quantum groups in two-dimensional physics, 1996.
DOI : 10.1017/CBO9780511628825

F. Goodman, P. De-la-harpe, and V. Jones, Coxeter graphs and towers of algebras, 1989.
DOI : 10.1007/978-1-4613-9641-3

V. Gorbounov, A. Isaev, and O. Ogievetsky, BRST Operator for Quantum Lie Algebras: Relation to the Bar Complex, Theoretical and Mathematical Physics, vol.139, issue.1, pp.473-485, 2004.
DOI : 10.1023/B:TAMP.0000022740.21580.d4

J. Grime, The hook fusion procedure, Electron, ) (2005) R26. ArXiv, p.502521

J. Grime, The hook fusion procedure for Hecke algebras, Journal of Algebra, vol.309, issue.2, pp.744-759, 2007.
DOI : 10.1016/j.jalgebra.2006.06.024

I. Grojnowski and M. Vazirani, Strong multiplicity one theorem for affine Hecke algebras of type A, Transf. Groups, pp.143-155, 2001.

V. Guizzi and P. Papi, A Combinatorial Approach to the Fusion Process for the Symmetric Group, European Journal of Combinatorics, vol.19, issue.7, pp.835-845, 1998.
DOI : 10.1006/eujc.1998.0242

P. Hoefsmit, Representations of Hecke algebras of finite groups with BN-pairs of classical type, 1974.

J. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, p.29, 1990.

S. Ihara and T. Yokonuma, On the second cohomology groups (Schur multipliers) of finite reflection groups, J. Fac. Sci. Univ. Tokyo Sect. I, pp.155-171, 1965.

A. Isaev, Quantum groups and Yang?Baxter equations, MPIM preprint, pp.2004-132

A. Isaev and A. Molev, Fusion procedure for the Brauer algebra, Algebra i Analiz, pp.142-154, 2010.

A. Isaev, A. Molev, and A. Os-'kin, On the Idempotents of Hecke Algebras, Letters in Mathematical Physics, vol.25, issue.1, pp.79-90, 2008.
DOI : 10.1007/s11005-008-0254-7

A. Isaev, A. Molev, and O. Ogievetsky, A New Fusion Procedure for the Brauer Algebra and Evaluation Homomorphisms, International Mathematics Research Notices, 2011.
DOI : 10.1093/imrn/rnr126
URL : https://hal.archives-ouvertes.fr/hal-00961354

A. Isaev, A. Molev, and O. Ogievetsky, Idempotents for Birman-Murakami-Wenzl algebras and reflection equation, Advances in Theoretical and Mathematical Physics, vol.18, issue.1, pp.1111-2502
DOI : 10.4310/ATMP.2014.v18.n1.a1
URL : https://hal.archives-ouvertes.fr/hal-01114891

A. Isaev and O. Ogievetsky, On representations of Hecke algebras, Czechoslovak Journal of Physics, vol.1, issue.11, pp.1433-1441, 2005.
DOI : 10.1007/s10582-006-0022-9

A. Isaev and O. Ogievetsky, On baxterized solutions of reflection equation and integrable chain models, Nuclear Physics B, vol.760, issue.3, pp.167-183, 2007.
DOI : 10.1016/j.nuclphysb.2006.09.013

A. Isaev and O. Ogievetsky, Jucys?Murphy elements for Birman-Murakami-Wenzl algebras, Physics of Particles and Nuclei Letters 8 No, pp.394-407, 2011.

A. Isaev and O. Ogievetsky, BRST OPERATOR FOR QUANTUM LIE ALGEBRAS: EXPLICIT FORMULA, International Journal of Modern Physics A, vol.19, issue.supp02, pp.19-240, 2004.
DOI : 10.1142/S0217751X04020440

A. Isaev, O. Ogievetsky, and P. Pyatov, On quantum matrix algebras satisfying the Cayley - Hamilton - Newton identities, Journal of Physics A: Mathematical and General, vol.32, issue.9, pp.115-1219809170, 1999.
DOI : 10.1088/0305-4470/32/9/002
URL : https://hal.archives-ouvertes.fr/hal-00473321

N. Iwahori and H. Matsumoto, On some bruhat decomposition and the structure of the hecke rings of p-Adic chevalley groups, Publications math??matiques de l'IH??S, vol.254, issue.1, pp.81-116, 1972.
DOI : 10.1007/BF02684396

V. Jain and O. Ogievetsky, Classical isomorphisms for quantum groups, Modern Phys, Lett. A, vol.7, issue.24, pp.2199-2209, 1992.

M. Jimbo, Aq-difference analogue of U(g) and the Yang-Baxter equation, Letters in Mathematical Physics, vol.5, issue.1, pp.63-69, 1985.
DOI : 10.1007/BF00704588

M. Jimbo, Introduction to the Yang?Baxter equation Braid group, knot theory and statistical mechanics, 1989.

M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, TheA n /(1) face models, Communications in Mathematical Physics, vol.6, issue.1, pp.543-565, 1988.
DOI : 10.1007/BF01218344

A. Jones, The Structure of the Young Symmetrizers for Spin Representations of the Symmetric Group, I, Journal of Algebra, vol.205, issue.2, pp.626-660, 1998.
DOI : 10.1006/jabr.1997.7400

V. Jones, Knots, braids and statistical mechanicsIntegrable systems and quantum groups, 1992.

A. Jones and M. Nazarov, -Analogues of the Young Symmetrizers for Projective Representations of the Symmetric Group, Proceedings of the London Mathematical Society, vol.78, issue.3, pp.481-512, 1999.
DOI : 10.1112/S002461159900177X
URL : https://hal.archives-ouvertes.fr/hal-00458910

A. Jucys, On the Young operators of the symmetric group, Lietuvos Fizikos Rinkinys, vol.6, pp.163-180, 1966.

C. Kassel, Quantum groups, 1995.
DOI : 10.1007/978-1-4612-0783-2
URL : https://hal.archives-ouvertes.fr/hal-00124690

A. Kleshchev, Linear and projective representations of symmetric groups, 2005.
DOI : 10.1017/CBO9780511542800

A. Klimyk and K. Schmüdgen, Quantum groups and their representations, 1998.
DOI : 10.1007/978-3-642-60896-4

P. Kulish, N. Reshetikhin, and E. Sklyanin, Yang?Baxter equation and representation theory : I, Lett, Math. Phys, vol.5, pp.393-403, 1981.

P. Kulish and E. Sklyanin, Quantum spectral transform method. Recent developments, Lecture Notes in Physics, pp.151-61, 1982.

P. Kulish and E. Sklyanin, Algebraic structures related to reflection equations, Journal of Physics A: Mathematical and General, vol.25, issue.22, pp.5963-5975, 1992.
DOI : 10.1088/0305-4470/25/22/022

S. Lambropoulou, Knot theory related to generalized and cyclotomic Hecke algebras of type ???, Journal of Knot Theory and Its Ramifications, vol.08, issue.05, pp.621-6580405504, 1999.
DOI : 10.1142/S0218216599000419

G. Lusztig, Hecke algebras with unequal parameters, CRM Monographs Ser, Amer. Math. Soc, vol.18, 2003.

I. Macdonald, Symmetric functions and Hall polynomials, 1998.

I. Macdonald, Affine Hecke Algebras and Orthogonal Polynomials, 2003.
DOI : 10.1017/CBO9780511542824

G. Malle and A. Mathas, Symmetric Cyclotomic Hecke Algebras, Journal of Algebra, vol.205, issue.1, pp.275-293, 1998.
DOI : 10.1006/jabr.1997.7339

G. Maxwell, The Schur multipliers of rotation subgroups of coxeter groups, Journal of Algebra, vol.53, issue.2, pp.440-451, 1978.
DOI : 10.1016/0021-8693(78)90290-9

L. Mezincescu and R. Nepomechie, Fusion procedure for open chains, Journal of Physics A: Mathematical and General, vol.25, issue.9, pp.2533-2543, 1992.
DOI : 10.1088/0305-4470/25/9/024

V. Miemietz, On Representations of Affine Hecke Algebras of Type B, Algebras and Representation Theory, vol.7, issue.3, pp.369-405, 2008.
DOI : 10.1007/s10468-008-9086-5

H. Mitsuhashi, The q-Analogue of the Alternating Group and Its Representations, Journal of Algebra, vol.240, issue.2, pp.535-5589912121, 2001.
DOI : 10.1006/jabr.2000.8715

H. Mitsuhashi, Z2-graded Clifford system in the Hecke algebra of type??Bn, Journal of Algebra, vol.264, issue.1, pp.231-250, 2003.
DOI : 10.1016/S0021-8693(03)00104-2

A. Molev, On the fusion procedure for the symmetric group, Reports on Mathematical Physics, vol.61, issue.2, pp.181-1880612207, 2008.
DOI : 10.1016/S0034-4877(08)80005-5

A. Morris, Projective Representations of Reflection Groups, Proceedings of the London Mathematical Society, vol.3, issue.3, pp.403-410, 1976.
DOI : 10.1112/plms/s3-32.3.403

G. E. Murphy, On the representation theory of the symmetric groups and associated Hecke algebras, Journal of Algebra, vol.152, issue.2, pp.492-513, 1992.
DOI : 10.1016/0021-8693(92)90045-N

M. Nazarov, Young's Symmetrizers for Projective Representations of the Symmetric Group, Advances in Mathematics, vol.127, issue.2, pp.190-257, 1997.
DOI : 10.1006/aima.1997.1621

M. Nazarov, C. Yangians, . I. Identitiesg, and . Olshanski, Kirillov's Seminar on Representation Theory ArXiv : q-alg, Math. Soc. Transl, vol.181, pp.139-163, 1998.

M. Nazarov, Mixed hook-length formula for degenerate a fine Hecke algebras, Lecture Notes in Math, vol.1815, pp.223-2369906148, 2003.
DOI : 10.1007/3-540-44890-X_10

M. Nazarov, A mixed hook-length formula for affine Hecke algebras, European Journal of Combinatorics, vol.25, issue.8, pp.1345-13760307091, 2004.
DOI : 10.1016/j.ejc.2003.10.010

M. Nazarov and V. Tarasov, On irreducibility of tensor products of Yangian modules associated with skew Young diagrams, Duke Math ArXiv : math, J, vol.112, pp.343-3780012039, 2002.

O. Ogievetsky, Uses of quantum spaces, Contemp. Math, vol.294, pp.161-232, 2002.
DOI : 10.1090/conm/294/04973
URL : https://hal.archives-ouvertes.fr/cel-00374419

O. Ogievetsky and P. Pyatov, Lecture on Hecke algebras, Proc of the Int. School "Symmetries and Integrable Systems, 1999.

O. Ogievetsky, W. B. Schmidke, J. Wess, and B. Zumino, q-Deformed Poincar?? algebra, Communications in Mathematical Physics, vol.1, issue.3, pp.495-518, 1992.
DOI : 10.1007/BF02096958

A. Okounkov and A. Vershik, A new approach to representation theory of symmetric groups II, Selecta Math (New series) 2 No ArXiv : math, pp.581-6050503040, 1996.

I. A. Pushkarev, On the representation theory of wreath products of finite groups and symmetric groups, Journal of Mathematical Sciences, vol.5, issue.5, pp.3590-3599, 1999.
DOI : 10.1007/BF02175835

A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras ArXiv :math, Proc. London Math. Soc. 75, pp.99-133, 1997.

A. Ram and A. Shepler, Classification of graded Hecke algebras for complex reflection groups, Commentarii Mathematici Helvetici, vol.78, issue.2, pp.308-3340209135, 2003.
DOI : 10.1007/s000140300013

L. Ratliff, The alternating Hecke algebra and its representations, 2007.

O. Schreier, Die Untergruppen der freien Gruppen, Abhandlungen aus dem Mathematischen Seminar der Universit??t Hamburg, vol.5, issue.1, pp.161-183, 1927.
DOI : 10.1007/BF02952517

J. D. Rogawski, On modules over the Hecke algebra of ap-adic group, Inventiones Mathematicae, vol.15, issue.No. 2, pp.443-465, 1985.
DOI : 10.1007/BF01388516

I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math, vol.139, pp.155-250, 1911.

G. Shephard and J. Todd, Finite unitary reflection groups, Canad, J. Math, vol.6, pp.274-304, 1954.

A. Sergeev and A. Vershik, A new approach to representation theory of symmetric groups. IV : Z 2 graded groups and algebras, Mosc. Math. J, vol.8, issue.4, pp.813-842, 2008.

W. Specht, Eine Verallgemeinerung der symmetrischen Gruppe, Schriften Math, Seminar (Berlin), vol.1, pp.1-32, 1932.

M. Vazirani, Parameterizing Hecke Algebra Modules: Bernstein-Zelevinsky Multisegments, Kleshchev Multipartitions, and Crystal Graphs, Transformation Groups, vol.7, issue.3, pp.267-303, 2002.
DOI : 10.1007/s00031-002-0014-1

A. Vershik, Local stationary algebras, Amer, Math. Soc. Trans, vol.148, issue.2, pp.1-13, 1991.

A. Vershik and M. Vsemirnov, The local stationary presentation of the alternating groups and the normal form, Journal of Algebra, vol.319, issue.10, pp.4222-42290703278, 2008.
DOI : 10.1016/j.jalgebra.2007.12.026

J. Wan, Wreath Hecke algebras and centralizer construction for wreath products, Journal of Algebra, vol.323, issue.9, pp.2371-2397, 2010.
DOI : 10.1016/j.jalgebra.2010.02.020

J. Wan and W. Wang, Modular Representations and Branching Rules for Wreath Hecke Algebras, International Mathematics Research Notices, pp.806-0196, 2008.
DOI : 10.1093/imrn/rnn128

W. Wang, Vertex algebras and the class algebras of wreath products, Proc. London Math, pp.381-404, 2004.
DOI : 10.1112/S0024611503014382

T. Yokonuma, On the second cohomology groups (Schur multipliers) of infinite discrete reflection groups, J. Fac. Sci. Univ. Tokyo Sect. I, pp.173-186, 1965.

A. Young, On quantitative substitutional analysis, Proc. London Math. Soc, pp.31-273, 1930.

R. Yue, Integrable high-spin chain related to the elliptic solution of the Yang-Baxter equation, Journal of Physics A: Mathematical and General, vol.27, issue.5, pp.1633-1644, 1994.
DOI : 10.1088/0305-4470/27/5/027

A. Zamolodchikov and Z. Al, Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models, Ann. Phys, vol.120, issue.253, 1979.

A. Zelevinsky, Induced representations of reductive ${\germ p}$-adic groups. II. On irreducible representations of ${\rm GL}(n)$, Annales scientifiques de l'??cole normale sup??rieure, vol.13, issue.2, pp.165-210, 1980.
DOI : 10.24033/asens.1379

P. Zinn-justin, Jucys???Murphy Elements and Weingarten Matrices, Letters in Mathematical Physics, vol.19, issue.5, pp.119-127, 2010.
DOI : 10.1007/s11005-009-0365-9