Generalized representation stability and 𝐹𝐼_{𝑑}-modules

Ramos, Eric

doi:10.1090/proc/13618

Generalized representation stability and $\mathrm {FI}_d$-modules
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Proc. Amer. Math. Soc. 145 (2017), 4647-4660 Request permission

Abstract:

In this note we consider the complex representation theory of $\mathrm {FI}_d$, a natural generalization of the category $\mathrm {FI}$ of finite sets and injections. We prove that finitely generated $\mathrm {FI}_d$-modules exhibit behaviors in the spirit of Church-Farb representation stability theory, generalizing a theorem of Church, Ellenberg, and Farb which connects finite generation of $\mathrm {FI}$-modules to representation stability.

References

T. Church and J. S. Ellenberg, Homology of FI-modules, arXiv:1506.01022.
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, 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
Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli, Representation theory of the symmetric groups, Cambridge Studies in Advanced Mathematics, vol. 121, Cambridge University Press, Cambridge, 2010. The Okounkov-Vershik approach, character formulas, and partition algebras. MR 2643487, DOI 10.1017/CBO9781139192361
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
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
Liping Li and Nina Yu, Filtrations and homological degrees of FI-modules, J. Algebra 472 (2017), 369–398. MR 3584882, DOI 10.1016/j.jalgebra.2016.11.019
Rohit Nagpal, FI-modules and the cohomology of modular representations of symmetric groups, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–The University of Wisconsin - Madison. MR 3358218
E. Ramos, Homological invariants of $\mathrm {FI}$-modules and $\mathrm {FI}_G$-modules, arXiv:1511.03964.
Steven V. Sam, Ideals of bounded rank symmetric tensors are generated in bounded degree, Invent. Math. 207 (2017), no. 1, 1–21. MR 3592755, DOI 10.1007/s00222-016-0668-2
Andrew Snowden, Syzygies of Segre embeddings and $\Delta$-modules, Duke Math. J. 162 (2013), no. 2, 225–277. MR 3018955, DOI 10.1215/00127094-1962767
Steven V. Sam and Andrew Snowden, Gröbner methods for representations of combinatorial categories, J. Amer. Math. Soc. 30 (2017), no. 1, 159–203. MR 3556290, DOI 10.1090/jams/859
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
S. Sam and A. Snowden, $GL$-equivariant modules over polynomial rings in infinitely many variables. II, in preparation.
Jennifer C. H. Wilson, $\rm FI_\scr W$-modules and stability criteria for representations of classical Weyl groups, J. Algebra 420 (2014), 269–332. MR 3261463, DOI 10.1016/j.jalgebra.2014.08.010