Representation stability for sequences of 0-Hecke modules

Laudone, Robert P.

doi:10.5802/alco.172

Laudone, Robert P.¹

¹ University of Michigan Dept. of mathematics 530 Church St. Ann Arbor MI 48109, USA

Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 619-661.

Abstract

We define a new category analogous to $FI$ for the $0$ -Hecke algebra $H_{n} (0)$ called the $0$ -Hecke category, $ℋ$ , indexing sequences of representations of $H_{n} (0)$ as $n$ varies under suitable compatibility conditions. We establish a new type of representation stability in this setting and prove it is implied by being a finitely generated $ℋ$ -module. We then provide examples of $ℋ$ -modules and discuss further desirable properties these modules possess.

Received: 2020-09-02
Revised: 2021-03-02
Accepted: 2021-03-08
Published online: 2021-09-02

DOI: 10.5802/alco.172

Classification: 13A50, 13C05, 16P40
Keywords: Representation stability, Diagram algebra, Hecke algebra

Author's affiliations:

Laudone, Robert P. ¹

¹ University of Michigan Dept. of mathematics 530 Church St. Ann Arbor MI 48109, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{ALCO_2021__4_4_619_0,
     author = {Laudone, Robert P.},
     title = {Representation stability for sequences of {0-Hecke} modules},
     journal = {Algebraic Combinatorics},
     pages = {619--661},
     publisher = {MathOA foundation},
     volume = {4},
     number = {4},
     year = {2021},
     doi = {10.5802/alco.172},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.172/}
}

TY  - JOUR
AU  - Laudone, Robert P.
TI  - Representation stability for sequences of 0-Hecke modules
JO  - Algebraic Combinatorics
PY  - 2021
SP  - 619
EP  - 661
VL  - 4
IS  - 4
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.172/
DO  - 10.5802/alco.172
LA  - en
ID  - ALCO_2021__4_4_619_0
ER  -

%0 Journal Article
%A Laudone, Robert P.
%T Representation stability for sequences of 0-Hecke modules
%J Algebraic Combinatorics
%D 2021
%P 619-661
%V 4
%N 4
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.172/
%R 10.5802/alco.172
%G en
%F ALCO_2021__4_4_619_0

Laudone, Robert P. Representation stability for sequences of 0-Hecke modules. Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 619-661. doi : 10.5802/alco.172. https://alco.centre-mersenne.org/articles/10.5802/alco.172/

References
Cited by

[1] Bigelow, Stephen Braid groups and Iwahori-Hecke algebras, Problems on mapping class groups and related topics (Proc. Sympos. Pure Math.), Volume 74, Amer. Math. Soc., Providence, RI, 2006, pp. 285-299 | DOI | MR | Zbl

[2] Brauer, Richard On algebras which are connected with the semisimple continuous groups, Ann. of Math. (2), Volume 38 (1937) no. 4, pp. 857-872 | DOI | MR | Zbl

[3] Church, Thomas; Ellenberg, Jordan S.; Farb, Benson FI-modules and stability for representations of symmetric groups, Duke Math. J., Volume 164 (2015) no. 9, pp. 1833-1910 | DOI | MR | Zbl

[4] Church, Thomas; Ellenberg, Jordan S.; Farb, Benson; Nagpal, Rohit FI-modules over Noetherian rings, Geom. Topol., Volume 18 (2014) no. 5, pp. 2951-2984 | DOI | MR | Zbl

[5] Draisma, Jan Noetherianity up to symmetry, Combinatorial algebraic geometry (Lecture Notes in Math.), Volume 2108, Springer, Cham, 2014, pp. 33-61 | DOI | MR | Zbl

[6] Draisma, Jan; Eggermont, Rob H. Plücker varieties and higher secants of Sato’s Grassmannian, J. Reine Angew. Math., Volume 737 (2018), pp. 189-215 | DOI | MR | Zbl

[7] Fulton, William; Harris, Joe Representation theory: a first course, Graduate Texts in Mathematics, 129, Springer Science & Business Media, 2013 | DOI | Zbl

[8] Gan, Wee Liang; Li, Liping An inductive machinery for representations of categories with shift functors, Trans. Amer. Math. Soc., Volume 371 (2019) no. 12, pp. 8513-8534 | DOI | MR | Zbl

[9] Geigle, Werner The Krull–Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences, Manuscripta Math., Volume 54 (1985) no. 1-2, pp. 83-106 | DOI | MR | Zbl

[10] Grinberg, Darij; Reiner, Victor Hopf algebras in combinatorics (2014) (https://arxiv.org/abs/1409.8356)

[11] Güntürkün, Sema; Snowden, Andrew The representation theory of the increasing monoid (2018) (https://arxiv.org/abs/1812.10242)

[12] Haglund, J.; Luoto, K.; Mason, S.; van Willigenburg, S. Quasisymmetric Schur functions, J. Combin. Theory Ser. A, Volume 118 (2011) no. 2, pp. 463-490 | DOI | MR | Zbl

[13] Halverson, Tom; Ram, Arun Bitraces for ${GL}_{n} (𝔽_{q})$ and the Iwahori–Hecke algebra of type $A_{n - 1}$ , Indag. Math. (N.S.), Volume 10 (1999) no. 2, pp. 247-268 | DOI | MR | Zbl

[14] Huang, Jia 0-Hecke algebra action on the Stanley–Reisner ring of the Boolean algebra, Ann. Comb., Volume 19 (2015) no. 2, pp. 293-323 | DOI | MR | Zbl

[15] Huang, Jia; Rhoades, Brendon Ordered set partitions and the 0-Hecke algebra, Algebr. Comb., Volume 1 (2018) no. 1, pp. 47-80 | DOI | Numdam | MR

[16] Jones, Vaughan F. R. The Potts model and the symmetric group, Subfactors (Kyuzeso, 1993), World Sci. Publ., River Edge, NJ, 1994, pp. 259-267 | MR | Zbl

[17] Koike, Kazuhiko On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters, Adv. Math., Volume 74 (1989) no. 1, pp. 57-86 | DOI | MR | Zbl

[18] Krob, Daniel; Thibon, Jean-Yves Noncommutative symmetric functions. IV. Quantum linear groups and Hecke algebras at $q = 0$ , J. Algebraic Combin., Volume 6 (1997) no. 4, pp. 339-376 | DOI | MR | Zbl

[19] Lam, Thomas; Pylyavskyy, Pavlo $P$ -partition products and fundamental quasi-symmetric function positivity, Adv. in Appl. Math., Volume 40 (2008) no. 3, pp. 271-294 | DOI | MR | Zbl

[20] Laudone, Robert P. Syzygies of secant ideals of Plücker-embedded Grassmannians are generated in bounded degree (2018) (https://arxiv.org/abs/1803.04259)

[21] Laudone, Robert P. The spin-Brauer diagram algebra, J. Algebraic Combin., Volume 50 (2019) no. 2, pp. 191-224 | DOI | MR | Zbl

[22] Lee, Min Ho Hecke operators on cohomology, Rev. Un. Mat. Argentina, Volume 50 (2009) no. 1, pp. 99-144 | MR | Zbl

[23] Macdonald, Ian G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages (With contributions by A. Zelevinsky, Oxford Science Publications) | MR

[24] Martin, Paul P. The partition algebra and the Potts model transfer matrix spectrum in high dimensions, J. Phys. A, Volume 33 (2000) no. 19, pp. 3669-3695 | DOI | MR | Zbl

[25] Mason, Sarah K. Recent trends in quasisymmetric functions, Recent trends in algebraic combinatorics (Assoc. Women Math. Ser.), Volume 16, Springer, Cham, 2019, pp. 239-279 | DOI | MR | Zbl

[26] Nagpal, Rohit; Sam, Steven V.; Snowden, Andrew Noetherianity of some degree two twisted commutative algebras, Selecta Math. (N.S.), Volume 22 (2016) no. 2, pp. 913-937 | DOI | MR | Zbl

[27] Norton, Pamela N. $0$ -Hecke algebras, J. Austral. Math. Soc. Ser. A, Volume 27 (1979) no. 3, pp. 337-357 | DOI | MR

[28] Patzt, Peter Representation stability for diagram algebras (2020) (https://arxiv.org/abs/2009.06346)

[29] Putman, Andrew; Sam, Steven V.; Snowden, Andrew Stability in the homology of unipotent groups, Algebra Number Theory, Volume 14 (2020) no. 1, pp. 119-154 | DOI | MR | Zbl

[30] Quillen, Daniel On the cohomology and $K$ -theory of the general linear groups over a finite field, Ann. of Math. (2), Volume 96 (1972), pp. 552-586 | DOI | MR | Zbl

[31] Ramos, Eric On the degree-wise coherence of ${FI}_{G}$ -modules, New York Journal of Mathematics, Volume 23 (2017), pp. 873-895 | MR | Zbl

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

[33] Rhie, Young Ho; Whaples, George Hecke operators in cohomology of groups, J. Math. Soc. Japan, Volume 22 (1970), pp. 431-442 | DOI | MR | Zbl

[34] Sam, Steven V. Ideals of bounded rank symmetric tensors are generated in bounded degree, Invent. Math., Volume 207 (2017) no. 1, pp. 1-21 | DOI | MR | Zbl

[35] Sam, Steven V.; Snowden, Andrew Introduction to twisted commutative algebras (2012) (https://arxiv.org/abs/1209.5122)

[36] Sam, Steven V.; Snowden, Andrew Stability patterns in representation theory, Forum Math. Sigma, Volume 3 (2015), Paper no. e11, 108 pages | DOI | MR | Zbl

[37] Sam, Steven V.; Snowden, Andrew GL-equivariant modules over polynomial rings in infinitely many variables, Trans. Amer. Math. Soc., Volume 368 (2016) no. 2, pp. 1097-1158 | DOI | MR | Zbl

[38] Sam, Steven V.; Snowden, Andrew Gröbner methods for representations of combinatorial categories, J. Amer. Math. Soc., Volume 30 (2017) no. 1, pp. 159-203 | DOI | MR | Zbl

[39] Sam, Steven V.; Snowden, Andrew The representation theory of Brauer categories I: triangular categories (2020) (https://arxiv.org/abs/2006.04328)

[40] Snowden, Andrew Syzygies of Segre embeddings and $Δ$ -modules, Duke Math. J., Volume 162 (2013) no. 2, pp. 225-277 | DOI | MR | Zbl

[41] Tewari, Vasu V.; van Willigenburg, Stephanie J. Quasisymmetric Schur functions and modules of the 0-Hecke algebra, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (Discrete Math. Theor. Comput. Sci. Proc., AT), Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2014, pp. 113-124 | MR | Zbl

[42] Tosteson, Philip Stability in the homology of Deligne–Mumford compactifications (2018) (arXiv preprint https://arxiv.org/abs/1801.03894)

[43] Turaev, Vladimir G. Operator invariants of tangles, and R-matrices, Math. USSR, Izv., Volume 35 (1990) no. 2, pp. 411-444 | DOI | MR | Zbl

Cited by Sources: