On the representation theory of an algebra of braids and ties
Steen Ryom-Hansen
DOI: 10.1007/s10801-010-0233-9
Abstract
We consider the algebra \Cal E n ( u) introduced by Aicardi and Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for \Cal E n ( u) and show that this is faithful. We use it to give a basis of \Cal E n ( u) and to classify its irreducible representations.
Pages: 57–79
Keywords: keywords diagram algebras; symmetric group; Specht modules
Full Text: PDF
References
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2. Ariki, S., Terasoma, T., Yamada, H.: Schur-Weyl reciprocity for the Hecke algebra of Z/rZ Sn. J. Algebra 178, 374-390 (1995)
3. Cheng, Y., Ge, M.L., Wu, Y.S., Xue, K.: Yang-Baxterization of braid group representations. Commun. Math. Phys. 136, 195-208 (1991)
4. Dipper, R., James, G.D.: Representation of Hecke algebras of general linear groups. Proc. Lond. Math. Soc. 54(3), 20-52 (1987)
5. Graham, J., Lehrer, G.I.: Cellular algebras. Invent. Math. 123, 1-34 (1996)
6. Halverson, T., Ram, A.: Partition algebras. Eur. J. Comb. 26, 869-921 (2005)
7. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29 (1990)
8. Goodman, F.M., de la Harpe, P., Jones, V.F.R.: Coxeter Graphs and Towers of Algebras. Mathematical Sciences Research Institute Publications, vol.
14. Springer, New York (1989)
9. Gyoja, A.: A q-analogue of Young symmetrizer. Osaka J. Math. 23, 841-852 (1986)
10. James, G.D.: The Representation Theory of the Symmetric Groups. Lecture Notes in Math., vol.
682. Springer, Berlin (1978)
11. Jones, V.F.R.: On a certain value of the Kauffman polynomial. Commun. Math. Phys. 125, 459 (1989)
12. Jones, V.F.R.: The Potts model and the symmetric group. In: Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebras, Kyuzeso, 1993, pp. 259-267. World Scientific, River Edge (1994)
13. Jimbo, M.: A q-difference analogue of U (g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63-69 (1985)
14. Juyumaya, J.: A new algebra arising from the representation theory of finite groups. Rev. Math. Phys. 11, 929-945 (1999)
15. Juyumaya, J.: Sur les nouveaux générateurs de l'algèbre de Hecke H(G, U, 1). J. Algebra 204, 40-68 (1998)
16. Juyumaya, J., Kennan, S.: Braid relations in the Yokonuma-Hecke algebra. J. Algebra 239, 272-295 (2001)
17. Martin, P.P.: Temperley-Lieb algebras for non-planar statistical mechanics-the partition algebra construction. J. Knot Theory Ramif. 3, 51-82 (1994)
18. Murphy, G.E.: On the representation theory of the symmetric group and associated Hecke algebras. J. Algebra 152, 492-519 (1992)
19. Ryom-Hansen, S.: The Ariki-Terasoma-Yamada tensor space and the blob algebra. J. Algebra, to appear
20. Thiem, N.: Unipotent Hecke algebras of Gln(Fq ). J. Algebra 294, 559-577 (2005)
21. Yokonuma, T.: Sur la structure des anneaux de Hecke d'un groupe de Chevalley fini. C. R. Acad. Sci.
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