The Long–Moody construction and polynomial functors
Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 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:
Accepted:
Published online:
DOI: 10.5802/aif.3282
Classification: 18A25, 18D10, 20C99, 20F36, 20J99
Keywords: braid groups, functor categories, Long–Moody construction, polynomial functors.
Mot clés : groupes de tresses, catégories de foncteurs, construction de Long–Moody, foncteurs polynomiaux
Soulié, Arthur 1

1 University of Strasbourg Institut de Recherche Mathématique Avancée 7 rue René Descartes 67084 Strasbourg (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2019__69_4_1799_0,
     author = {Souli\'e, Arthur},
     title = {The {Long{\textendash}Moody} construction and polynomial functors},
     journal = {Annales de l'Institut Fourier},
     pages = {1799--1856},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {4},
     year = {2019},
     doi = {10.5802/aif.3282},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3282/}
}
TY  - JOUR
AU  - Soulié, Arthur
TI  - The Long–Moody construction and polynomial functors
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 1799
EP  - 1856
VL  - 69
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3282/
DO  - 10.5802/aif.3282
LA  - en
ID  - AIF_2019__69_4_1799_0
ER  - 
%0 Journal Article
%A Soulié, Arthur
%T The Long–Moody construction and polynomial functors
%J Annales de l'Institut Fourier
%D 2019
%P 1799-1856
%V 69
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3282/
%R 10.5802/aif.3282
%G en
%F AIF_2019__69_4_1799_0
Soulié, Arthur. 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/articles/10.5802/aif.3282/

[1] Bigelow, Stephen The Lawrence-Krammer representation, Topology and geometry of manifolds (Athens, GA, 2001) (Proceedings of Symposia in Pure Mathematics), Volume 71, American Mathematical Society, 2003, pp. 51-68 | DOI | MR | Zbl

[2] Bigelow, Stephen Homological representations of the Iwahori–Hecke algebra, Proceedings of the Casson Fest (Geometry and Topology Monographs), Volume 7 (2004), pp. 493-507 | DOI | MR | Zbl

[3] Bigelow, Stephen; Tian, Jianjun Paul Generalized Long–Moody representations of braid groups, Commun. Contemp. Math., Volume 10 (2008) no. suppl. 1, pp. 1093-1102 | DOI | MR | Zbl

[4] Birman, Joan S. Braids, links, and mapping class groups, 82, Princeton University Press; University of Tokyo Press, 1974, ix+228 pages (Annals of Mathematics Studies) | MR | Zbl

[5] Birman, Joan S.; Brendle, Tara E. Braids: a survey, Handbook of knot theory, Elsevier, 2005, pp. 19-103 | DOI | MR | Zbl

[6] Cohen, Daniel E. Groups of cohomological dimension one, Lecture Notes in Mathematics, 245, Springer, 1972, v+99 pages | MR | Zbl

[7] Djament, Aurélien; Vespa, Christine Foncteurs faiblement polynomiaux, Int. Math. Res. Not. (2019) no. 2, pp. 321-391 | DOI | MR | Zbl

[8] Eilenberg, Samuel; Mac Lane, Saunders On the groups H(Π,n). II. Methods of computation, Ann. Math., Volume 60 (1954), pp. 49-139 | DOI | MR | Zbl

[9] Grayson, Daniel Higher algebraic K-theory. II (after Daniel Quillen), Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, IL, 1976) (Lecture Notes in Mathematics), Volume 551, Springer, 1976, pp. 217-240 | MR | Zbl

[10] Ito, Tetsuya The classification of Wada-type representations of braid groups, J. Pure Appl. Algebra, Volume 217 (2013) no. 9, pp. 1754-1763 | DOI | MR | Zbl

[11] van der Kallen, Wilberd Homology stability for linear groups, Invent. Math., Volume 60 (1980) no. 3, pp. 269-295 | DOI | MR | Zbl

[12] Kassel, Christian; Turaev, Vladimir Braid groups, Graduate Texts in Mathematics, 247, Springer, 2008, xii+340 pages (with the graphical assistance of Olivier Dodane) | DOI | MR | Zbl

[13] Kohno, Toshitake Homological representations of braid groups and KZ connections, J. Singul., Volume 5 (2012), pp. 94-108 | MR | Zbl

[14] Krammer, Daan Braid groups are linear, Ann. Math., Volume 155 (2002) no. 1, pp. 131-156 | DOI | MR | Zbl

[15] Lawrence, Ruth J. A topological approach to representations of the Iwahori–Hecke algebra, Int. J. Mod. Phys. A, Volume 5 (1990) no. 16, pp. 3213-3219 | DOI | MR | Zbl

[16] Long, Darren D. On the linear representation of braid groups, Trans. Am. Math. Soc., Volume 311 (1989) no. 2, pp. 535-560 | DOI | MR | Zbl

[17] Long, Darren D. On the linear representation of braid groups. II, Duke Math. J., Volume 59 (1989) no. 2, pp. 443-460 | DOI | MR

[18] Long, Darren D. Constructing representations of braid groups, Commun. Anal. Geom., Volume 2 (1994) no. 2, pp. 217-238 | DOI | MR | Zbl

[19] Mac Lane, Saunders Categories for the working mathematician, Graduate Texts in Mathematics, 5, Springer, 1998, xii+314 pages | MR | Zbl

[20] Marin, Ivan On the representation theory of braid groups, Ann. Math. Blaise Pascal, Volume 20 (2013) no. 2, pp. 193-260 | DOI | MR | Zbl

[21] Randal-Williams, Oscar; Wahl, Nathalie Homological stability for automorphism groups, Adv. Math., Volume 318 (2017), pp. 534-626 | DOI | MR | Zbl

[22] Soulié, Arthur Generalized Long–Moody functors (2018) (https://arxiv.org/abs/1709.04278)

[23] Tong, Dian-Min; Yang, Shan-De; Ma, Zhong-Qi A new class of representations of braid groups, Commun. Theor. Phys., Volume 26 (1996) no. 4, pp. 483-486 | DOI | MR

[24] Wada, Masaaki Group invariants of links, Topology, Volume 31 (1992) no. 2, pp. 399-406 | DOI | MR | Zbl

[25] Weibel, Charles A. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, 1994, xiv+450 pages | DOI | MR | Zbl

Cited by Sources: