The Long–Moody construction and polynomial functors
Annales de l'Institut Fourier, Volume 69 (2019) no. 4, p. 1799-1856

In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of 𝔹 n with a representation of 𝔹 n+1 . In this paper, we prove that this construction is functorial and can be extended: it inspires endofunctors, called Long–Moody functors, on the category of functors from Quillen’s bracket construction associated with the braid groupoid to a module category. Then we study the effect of Long–Moody functors on strong polynomial functors: we prove that they increase by one the degree of very strong polynomiality.

En 1994, Long et Moody ont donné une construction sur les représentations des groupes de tresses, associant une représentation de 𝔹 n à une représentation de 𝔹 n+1 . Dans cet article, on démontre que cette construction est fonctorielle et qu’elle peut s’étendre : elle est à l’origine d’endofoncteur, appelés endofoncteurs de Long–Moody, sur la catégorie des foncteurs ayant une construction de Quillen pour catégorie source et une catégorie de modules pour but. Ensuite, nous étudions l’effet des foncteurs de Long–Moody sur les foncteurs fortement polynomiaux : on démontre qu’ils augmentent de un le degré de très forte polynomialité.

Received : 2018-04-26
Accepted : 2018-09-25
Published online : 2019-09-16
DOI : https://doi.org/10.5802/aif.3282
Classification:  18A25,  18D10,  20C99,  20F36,  20J99
Keywords: braid groups, functor categories, Long–Moody construction, polynomial functors.
@article{AIF_2019__69_4_1799_0,
     author = {Souli\'e, Arthur},
     title = {The Long--Moody construction and polynomial functors},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {4},
     year = {2019},
     pages = {1799-1856},
     doi = {10.5802/aif.3282},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_4_1799_0}
}
The Long–Moody construction and polynomial functors. Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 1799-1856. doi : 10.5802/aif.3282. https://aif.centre-mersenne.org/item/AIF_2019__69_4_1799_0/

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