Some examples of simple generic 𝐹𝐼-modules in positive characteristic

Kriz, Sophie

doi:10.1090/ert/666

Some examples of simple generic $FI$-modules in positive characteristic
HTML articles powered by AMS MathViewer

by Sophie Kriz;

Represent. Theory 27 (2023), 1194-1207

DOI: https://doi.org/10.1090/ert/666

Published electronically: November 30, 2023

HTML | PDF | Request permission

Abstract:

We give, in any characteristic $p>0$, examples of simple generic $FI$-modules whose underlying representations are reducible in all sufficiently high degrees.

References

Thomas Church and Jordan S. Ellenberg, Homology of FI-modules, Geom. Topol. 21 (2017), no. 4, 2373–2418. MR 3654111, DOI 10.2140/gt.2017.21.2373
Thomas Church, Jordan S. Ellenberg, and Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833–1910. MR 3357185, DOI 10.1215/00127094-3120274
Thomas Church, Jordan S. Ellenberg, Benson Farb, and Rohit Nagpal, FI-modules over Noetherian rings, Geom. Topol. 18 (2014), no. 5, 2951–2984. MR 3285226, DOI 10.2140/gt.2014.18.2951
Thomas Church and Benson Farb, Representation theory and homological stability, Adv. Math. 245 (2013), 250–314. MR 3084430, DOI 10.1016/j.aim.2013.06.016
Thomas Church, Jeremy Miller, Rohit Nagpal, and Jens Reinhold, Linear and quadratic ranges in representation stability, Adv. Math. 333 (2018), 1–40. MR 3818071, DOI 10.1016/j.aim.2018.05.025
Matthew Fayers, Reducible Specht modules, J. Algebra 280 (2004), no. 2, 500–504. MR 2089249, DOI 10.1016/j.jalgebra.2003.09.053
Matthew Fayers, Irreducible Specht modules for Hecke algebras of type $\textbf {A}$, Adv. Math. 193 (2005), no. 2, 438–452. MR 2137291, DOI 10.1016/j.aim.2004.06.001
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821, DOI 10.24033/bsmf.1583
Nir Gadish, Categories of FI type: a unified approach to generalizing representation stability and character polynomials, J. Algebra 480 (2017), 450–486. MR 3633316, DOI 10.1016/j.jalgebra.2017.03.010
Wee Liang Gan and Liping Li, Coinduction functor in representation stability theory, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 689–711. MR 3431657, DOI 10.1112/jlms/jdv043
Nate Harman, Virtual Specht stability for $FI$-modules in positive characteristic, J. Algebra 488 (2017), 29–41. MR 3680911, DOI 10.1016/j.jalgebra.2017.06.006
G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828, DOI 10.1007/BFb0067708
G. D. James, On the decomposition matrices of the symmetric groups. II, J. Algebra 43 (1976), no. 1, 45–54. MR 430050, DOI 10.1016/0021-8693(76)90143-5
G. D. James, On the decomposition matrices of the symmetric groups. III, J. Algebra 71 (1981), no. 1, 115–122. MR 627427, DOI 10.1016/0021-8693(81)90108-3
Gordon James and Andrew Mathas, The irreducible Specht modules in characteristic $2$, Bull. London Math. Soc. 31 (1999), no. 4, 457–462. MR 1687552, DOI 10.1112/S0024609399005822
Sophie Kriz, On the local cohomology of L-shaped integral FI-modules, J. Algebra 611 (2022), 149–174. MR 4472427, DOI 10.1016/j.jalgebra.2022.08.003
Liping Li, Upper bounds of homological invariants of $FI_G$-modules, Arch. Math. (Basel) 107 (2016), no. 3, 201–211. MR 3538516, DOI 10.1007/s00013-016-0921-3
Liping Li and Eric Ramos, Depth and the local cohomology of $\mathcal {FI}_G$-modules, Adv. Math. 329 (2018), 704–741. MR 3783426, DOI 10.1016/j.aim.2018.02.029
Jeremy Miller and Jennifer C. H. Wilson, Quantitative representation stability over linear groups, Int. Math. Res. Not. IMRN 22 (2020), 8624–8672. MR 4216699, DOI 10.1093/imrn/rny250
Rohit Nagpal, VI-modules in nondescribing characteristic, part I, Algebra Number Theory 13 (2019), no. 9, 2151–2189. MR 4039499, DOI 10.2140/ant.2019.13.2151
Rohit Nagpal, VI-modules in non-describing characteristic, part II, J. Reine Angew. Math. 781 (2021), 187–205. MR 4343100, DOI 10.1515/crelle-2021-0054
Andrew Putman and Steven V. Sam, Representation stability and finite linear groups, Duke Math. J. 166 (2017), no. 13, 2521–2598. MR 3703435, DOI 10.1215/00127094-2017-0008
Eric Ramos, Homological invariants of $\textrm {FI}$-modules and $\textrm {FI}_G$-modules, J. Algebra 502 (2018), 163–195. MR 3774889, DOI 10.1016/j.jalgebra.2017.12.037
Steven V. Sam and Andrew Snowden, GL-equivariant modules over polynomial rings in infinitely many variables, Trans. Amer. Math. Soc. 368 (2016), no. 2, 1097–1158. MR 3430359, DOI 10.1090/tran/6355
Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278 (French). MR 68874, DOI 10.2307/1969915