Algebras Defined by Reflections

Hecke algebra を始めとして, reflection からは様々な代数が定義される。 Coxeter group (Coxeter system) や complex reflection group など, reflection group 自体様々な一般化や変種があるので, それらの代数も対応したものが考えられている。

まず Hecke algebra の変種に様々なものがあるが, それについては, このページに挙げた。 それ以外に, 目にしたものを挙げると次のようになる。

• Schur algebra とその変種
• quasi-Coxeter algebra [Tol08]
• descent algebra [Sol76; Sal08]
• peak algebra [Nym03; ABN04]
• face semigroup algebra [Sal08]
• nil-Coxeter algebra [FS94]
• generalized nil-Coxeter algebra [Kha17; Kha18b; Kha18a]
• bracket algebra [KM04]
• Hecke group algebra [HT09]
• Temerley-Lieb algebra
• Fomin-Kirillov algebra [FK99; Hom]

References

[ABN04]

Marcelo Aguiar, Nantel Bergeron, and Kathryn Nyman. “The peak algebra and the descent algebras of types \(B\) and \(D\)”. In: Trans. Amer. Math. Soc. 356.7 (2004), pp. 2781–2824. arXiv: math/0302278. url: https://doi.org/10.1090/S0002-9947-04-03541-X.

[FK99]

Sergey Fomin and Anatol N. Kirillov. “Quadratic algebras, Dunkl elements, and Schubert calculus”. In: Advances in geometry. Vol. 172. Progr. Math. Birkhäuser Boston, Boston, MA, 1999, pp. 147–182.

[FS94]

Sergey Fomin and Richard P. Stanley. “Schubert polynomials and the nil-Coxeter algebra”. In: Adv. Math. 103.2 (1994), pp. 196–207. url: https://doi.org/10.1006/aima.1994.1009.

[Hom]

Sirous Homayouni. A quotient of Fomin-Kirillov Algebra and q-Lucas polynomial. arXiv: 2111.10982.

[HT09]

Florent Hivert and Nicolas M. Thiéry. “The Hecke group algebra of a Coxeter group and its representation theory”. In: J. Algebra 321.8 (2009), pp. 2230–2258. arXiv: 0711.1561. url: http://dx.doi.org/10.1016/j.jalgebra.2008.09.039.

[Kha17]

Apoorva Khare. “Generalized nil-Coxeter algebras, cocommutative algebras, and the PBW property”. In: Groups, rings, group rings, and Hopf algebras. Vol. 688. Contemp. Math. Amer. Math. Soc., Providence, RI, 2017, pp. 139–168. arXiv: 1601.04775. url: https://doi.org/10.1090/conm/688.

[Kha18a]

Apoorva Khare. “Generalized nil-Coxeter algebras”. In: Sém. Lothar. Combin. 80B (2018), Art. 29, 12. arXiv: 1802.07015.

[Kha18b]

Apoorva Khare. “Generalized nil-Coxeter algebras over discrete complex reflection groups”. In: Trans. Amer. Math. Soc. 370.4 (2018), pp. 2971–2999. arXiv: 1601.08231. url: https://doi.org/10.1090/tran/7304.

[KM04]

Anatol N. Kirillov and Toshiaki Maeno. “Noncommutative algebras related with Schubert calculus on Coxeter groups”. In: European J. Combin. 25.8 (2004), pp. 1301–1325. arXiv: math/0310068. url: http://dx.doi.org/10.1016/j.ejc.2003.11.006.

[Nym03]

Kathryn L. Nyman. “The peak algebra of the symmetric group”. In: J. Algebraic Combin. 17.3 (2003), pp. 309–322. url: https://doi.org/10.1023/A:1025000905826.

[Sal08]

Franco V. Saliola. “On the quiver of the descent algebra”. In: J. Algebra 320.11 (2008), pp. 3866–3894. arXiv: 0708.4213. url: http://dx.doi.org/10.1016/j.jalgebra.2008.07.009.

[Sol76]

Louis Solomon. “A Mackey formula in the group ring of a Coxeter group”. In: J. Algebra 41.2 (1976), pp. 255–264. url: https://doi.org/10.1016/0021-8693(76)90182-4.

[Tol08]

Valerio Toledano Laredo. “Quasi-Coxeter algebras, Dynkin diagram cohomology, and quantum Weyl groups”. In: Int. Math. Res. Pap. IMRP (2008), Art. ID rpn009, 167. arXiv: math/0506529.