Abstract
A celebrated result of Farahat and Higman constructs an algebra FH which “interpolates” the centres \(Z(\mathbb {Z}S_{n})\) of group algebras of the symmetric groups Sn. We extend these results from symmetric group algebras to type A Iwahori-Hecke algebras, Hn(q). In particular, we explain how to construct an algebra FHq “interpolating” the centres Z(Hn(q)). We prove that FHq is isomorphic to \(\mathcal {R}[q,q^{-1}] \otimes _{\mathbb {Z}} {\Lambda }\) (where \(\mathcal {R}\) is the ring of integer-valued polynomials, and Λ is the ring of symmetric functions). The isomorphism can be described as “evaluation at Jucys-Murphy elements”, leading to a proof of a conjecture of Francis and Wang. This yields character formulae for the Geck-Rouquier basis of Z(Hn(q)) when acting on Specht modules.
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References
Brauer, R., de B Robinson, G: On a conjecture by Nakayama. Trans. Roy. Soc. Canada. Sect. III(41), 20–25 (1947)
Corteel, S., Goupil, A., Schaeffer, G.: Content evaluation and class symmetric functions. Adv. Math. 188(2), 315–336 (2004)
Francis, A. R, Graham, J. J: Centres of Hecke algebras: the Dipper–James conjecture. J. Algebra 306(1), 244–267 (2006)
Farahat, H. K., Higman, G.: The centres of symmetric group rings. Proc. R. Soc. London. Series A Math. Phys. Sci. 250(1261), 212–221 (1959)
Francis, A., Wang, W.: The centers of Iwahori-Hecke algebras are filtered. Representation Theory, Comtemporary Mathematics 478, 29–38 (2009)
Geck, M., Rouquier, R.: Centers and simple modules for Iwahori-Hecke algebras. In: Finite Reductive Groups: Related Structures and Representations, pp. 251–272. Springer (1997)
Ivanov, V N, Kerov, S V: The algebra of conjugacy classes in symmetric groups and partial permutations. J. Math. Sci. 107(5), 4212–4230 (2001)
Jucys, A-AA: Symmetric polynomials and the center of the symmetric group ring. Rep. Math. Phys. 5(1), 107–112 (1974)
Kannan, AS, Ryba, C: Stable centres II: Finite classical groups. arXiv:2112.01467 (2021)
Méliot, P: Products of Geck-Rouquier conjugacy classes and the Hecke algebra of composed permutations. Discrete Mathematics & Theoretical Computer Science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010)
Macdonald, I.G.: With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008 paperback edition [MR1354144], 2nd edn. The Clarendon Press, Oxford University Press, New York (2015)
Andrew, M.: Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group Volume, vol. 15. American Mathematical Society Providence, RI (1999)
Andrew, M.: Murphy operators and the centre of the Iwahori-Hecke algebras of type A. J. Algebraic Combin. 9(3), 295–313 (1999)
Murphy, G.E: The idempotents of the symmetric group and Nakayama’s conjecture. J. Algebra 81(1), 258–265 (1983)
Christopher, R.: Stable centres I: Wreath products. arXiv:2107.03752 (2021)
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The author would like to thank Weiqiang Wang for helpful comments.
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Presented by: Andrew Mathas
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Ryba, C. Stable Centres of Iwahori-Hecke Algebras of Type A. Algebr Represent Theor 26, 2343–2359 (2023). https://doi.org/10.1007/s10468-022-10184-9
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DOI: https://doi.org/10.1007/s10468-022-10184-9