Gröbner methods for representations of combinatorial categories

Sam, Steven; Snowden, Andrew

doi:10.1090/jams/859

Gröbner methods for representations of combinatorial categories
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by Steven V Sam and Andrew Snowden

J. Amer. Math. Soc. 30 (2017), 159-203

DOI: https://doi.org/10.1090/jams/859

Published electronically: March 17, 2016

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Abstract:

Given a category $\mathcal {C}$ of a combinatorial nature, we study the following fundamental question: how do combinatorial properties of $\mathcal {C}$ affect algebraic properties of representations of $\mathcal {C}$? We prove two general results. The first gives a criterion for representations of $\mathcal {C}$ to admit a theory of Gröbner bases, from which we obtain a criterion for noetherianity. The second gives a criterion for a general “rationality” result for Hilbert series of representations of $\mathcal {C}$, and connects to the theory of formal languages.

Our work is motivated by recent work in the literature on representations of various specific categories. Our general criteria recover many of the results on these categories that had been proved by ad hoc means, and often yield cleaner proofs and stronger statements. For example, we give a new, more robust, proof that FI-modules (studied by Church, Ellenberg, and Farb), and certain generalizations, are noetherian; we prove the Lannes–Schwartz artinian conjecture from the study of generic representation theory of finite fields; we significantly improve the theory of $\Delta$-modules, introduced by Snowden in connection to syzygies of Segre embeddings; and we establish fundamental properties of twisted commutative algebras in positive characteristic.

References

M. G. Barratt, Twisted Lie algebras, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 9–15. MR 513566
Miklós Bóna, Combinatorics of permutations, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2012. With a foreword by Richard Stanley. MR 2919720, DOI 10.1201/b12210
N. Chomsky and M. P. Schützenberger, The algebraic theory of context-free languages, Computer programming and formal systems, North-Holland, Amsterdam, 1963, pp. 118–161. MR 0152391
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
D. E. Cohen, On the laws of a metabelian variety, J. Algebra 5 (1967), 267–273. MR 206104, DOI 10.1016/0021-8693(67)90039-7
Aurélien Djament, Foncteurs en grassmanniennes, filtration de Krull et cohomologie des foncteurs, Mém. Soc. Math. Fr. (N.S.) 111 (2007), xxii+213 pp. (2008) (French, with English and French summaries). MR 2482711, DOI 10.24033/msmf.423
Aurélien Djament, Le foncteur $V\mapsto \Bbb F_2[V]^{\otimes 3}$ entre $\Bbb F_2$-espaces vectoriels est noethérien, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 2, 459–490 (French, with English and French summaries). MR 2521424, DOI 10.5802/aif.2436
A. Djament, Des propriétés de finitude des foncteurs polynomiaux, available at arXiv:1308.4698v1., DOI 10.4064/fm954-8-2015
Aurélien Djament, La propriété noethérienne pour les foncteurs entre espaces vectoriels (d’après A. Putman, S. Sam et A. Snowden), Séminaire Bourbaki, Vol. 2014/2015, Astérisque, to appear.
Aurélien Djament and Christine Vespa, Sur l’homologie des groupes orthogonaux et symplectiques à coefficients tordus, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 3, 395–459 (French, with English and French summaries). MR 2667021, DOI 10.24033/asens.2125
Vladimir Dotsenko and Anton Khoroshkin, Gröbner bases for operads, Duke Math. J. 153 (2010), no. 2, 363–396. MR 2667136, DOI 10.1215/00127094-2010-026
Vladimir Dotsenko and Anton Khoroshkin, Shuffle algebras, homology, and consecutive pattern avoidance, Algebra Number Theory 7 (2013), no. 3, 673–700. MR 3095223, DOI 10.2140/ant.2013.7.673
Randall Dougherty, Functors on the category of finite sets, Trans. Amer. Math. Soc. 330 (1992), no. 2, 859–886. MR 1053111, DOI 10.1090/S0002-9947-1992-1053111-4
Jan Draisma, Noetherianity up to symmetry, Combinatorial algebraic geometry, Lecture Notes in Math., vol. 2108, Springer, Cham, 2014, pp. 33–61. MR 3329086, DOI 10.1007/978-3-319-04870-3_{2}
Jan Draisma and Rob H. Eggermont, Plücker varieties and higher secants of Sato’s Grassmannian, available at arXiv:1402.1667v2. J. Reine Angew. Math., to appear., DOI 10.1515/crelle-2015-0035
Jan Draisma and Jochen Kuttler, Bounded-rank tensors are defined in bounded degree, Duke Math. J. 163 (2014), no. 1, 35–63. MR 3161311, DOI 10.1215/00127094-2405170
Samuel Eilenberg and Saunders Mac Lane, On the groups $H(\Pi ,n)$. II. Methods of computation, Ann. of Math. (2) 60 (1954), 49–139. MR 65162, DOI 10.2307/1969702
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
Benson Farb, Representation stability, Proceedings of the International Congress of Mathematicians, Volume II, pp. 1173–1196, available at arXiv:1404.4065v1.
Vincent Franjou, Eric M. Friedlander, Alexander Scorichenko, and Andrei Suslin, General linear and functor cohomology over finite fields, Ann. of Math. (2) 150 (1999), no. 2, 663–728. MR 1726705, DOI 10.2307/121092
Vincent Franjou, Jean Lannes, and Lionel Schwartz, Autour de la cohomologie de Mac Lane des corps finis, Invent. Math. 115 (1994), no. 3, 513–538 (French, with English and French summaries). MR 1262942, DOI 10.1007/BF01231771
Eric M. Friedlander and Andrei Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209–270. MR 1427618, DOI 10.1007/s002220050119
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821, DOI 10.24033/bsmf.1583
Wee Liang Gan and Liping Li, Noetherian property of infinite EI categories, New York J. Math. 21 (2015), 369–382. MR 3358549
Scott Garrabrant and Igor Pak, Pattern avoidance is not P-recursive, available at arXiv:1505.06508v1.
Victor Ginzburg and Travis Schedler, Differential operators and BV structures in noncommutative geometry, Selecta Math. (N.S.) 16 (2010), no. 4, 673–730. MR 2734329, DOI 10.1007/s00029-010-0029-8
Mitsuyasu Hashimoto, Determinantal ideals without minimal free resolutions, Nagoya Math. J. 118 (1990), 203–216. MR 1060711, DOI 10.1017/S0027763000003081
Graham Higman, Ordering by divisibility in abstract algebras, Proc. London Math. Soc. (3) 2 (1952), 326–336. MR 49867, DOI 10.1112/plms/s3-2.1.326
Christopher J. Hillar and Abraham Martín del Campo, Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals, J. Symbolic Comput. 50 (2013), 314–334. MR 2996883, DOI 10.1016/j.jsc.2012.06.006
Christopher J. Hillar and Seth Sullivant, Finite Gröbner bases in infinite dimensional polynomial rings and applications, Adv. Math. 229 (2012), no. 1, 1–25. MR 2854168, DOI 10.1016/j.aim.2011.08.009
John E. Hopcroft and Jeffrey D. Ullman, Introduction to automata theory, languages, and computation, Addison-Wesley Series in Computer Science, Addison-Wesley Publishing Co., Reading, Mass., 1979. MR 645539
Anton Khoroshkin and Dmitri Piontkovski, On generating series of finitely presented operads, J. Algebra 426 (2015), 377–429. MR 3301915, DOI 10.1016/j.jalgebra.2014.12.012
Henning Krause, The artinian conjecture (following Djament, Putman, Sam, and Snowden), Proceedings of the 47th Symposium on Ring Theory and Representation Theory, Symp. Ring Theory Represent. Theory Organ. Comm., Okayama, 2015, pp. 104–111. MR 3363953
Nicholas J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. I, Amer. J. Math. 116 (1994), no. 2, 327–360. MR 1269607, DOI 10.2307/2374932
Nicholas J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. II, $K$-Theory 8 (1994), no. 4, 395–428. MR 1300547, DOI 10.1007/BF00961409
Nicholas J. Kuhn, Invariant subspaces of the ring of functions on a vector space over a finite field, J. Algebra 191 (1997), no. 1, 212–227. MR 1444496, DOI 10.1006/jabr.1996.6922
Nicholas J. Kuhn, The generic representation theory of finite fields: a survey of basic structure, Infinite length modules (Bielefeld, 1998) Trends Math., Birkhäuser, Basel, 2000, pp. 193–212. MR 1789216
Nicholas J. Kuhn, Generic representation theory of finite fields in nondescribing characteristic, Adv. Math. 272 (2015), 598–610. MR 3303242, DOI 10.1016/j.aim.2014.12.012
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Rohit Nagpal, Steven V Sam, and Andrew Snowden, Noetherianity of some degree two twisted commutative algebras, available at arXiv:1501.06925v2. Selecta Math. (N.S.), to appear.
C. St. J. A. Nash-Williams, On well-quasi-ordering finite trees, Proc. Cambridge Philos. Soc. 59 (1963), 833–835. MR 153601, DOI 10.1017/s0305004100003844
Luke Oeding and Claudiu Raicu, Tangential varieties of Segre-Veronese varieties, Collect. Math. 65 (2014), no. 3, 303–330. MR 3240996, DOI 10.1007/s13348-014-0111-1
Robert Paré, Contravariant functors on finite sets and Stirling numbers, Theory Appl. Categ. 6 (1999), 65–76. The Lambek Festschrift. MR 1732463
Teimuraz Pirashvili, Dold-Kan type theorem for $\Gamma$-groups, Math. Ann. 318 (2000), no. 2, 277–298. MR 1795563, DOI 10.1007/s002080000120
Teimuraz Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 2, 151–179 (English, with English and French summaries). MR 1755114, DOI 10.1016/S0012-9593(00)00107-5
Geoffrey M. L. Powell, The Artinian conjecture for $I^{\otimes 2}$, J. Pure Appl. Algebra 128 (1998), no. 3, 291–310. With an appendix by Lionel Schwartz. MR 1626361, DOI 10.1016/S0022-4049(97)00054-6
Geoffrey M. L. Powell, The structure of indecomposable injectives in generic representation theory, Trans. Amer. Math. Soc. 350 (1998), no. 10, 4167–4193. MR 1458333, DOI 10.1090/S0002-9947-98-02125-4
Geoffrey M. L. Powell, On Artinian objects in the category of functors between $\mathbf F_2$-vector spaces, Infinite length modules (Bielefeld, 1998) Trends Math., Birkhäuser, Basel, 2000, pp. 213–228. MR 1789217
Andrew Putman and Steven V Sam, Representation stability and finite linear groups, available at arXiv:1408.3694v2.
Claudiu Raicu, Secant varieties of Segre-Veronese varieties, Algebra Number Theory 6 (2012), no. 8, 1817–1868. MR 3033528, DOI 10.2140/ant.2012.6.1817
Birgit Richter, Taylor towers for $\Gamma$-modules, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 4, 995–1023 (English, with English and French summaries). MR 1849212, DOI 10.5802/aif.1842
Günther Richter, Noetherian semigroup rings with several objects, Group and semigroup rings (Johannesburg, 1985) North-Holland Math. Stud., vol. 126, North-Holland, Amsterdam, 1986, pp. 231–246. MR 860064
María Ronco, Shuffle bialgebras, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 3, 799–850 (English, with English and French summaries). MR 2918719, DOI 10.5802/aif.2630
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
Steven V Sam and Andrew Snowden, Introduction to twisted commutative algebras, available at arXiv:1209.5122v1.
Steven V. Sam and Andrew Snowden, Stability patterns in representation theory, Forum Math. Sigma 3 (2015), Paper No. e11, 108. MR 3376738, DOI 10.1017/fms.2015.10
Steven V Sam and Andrew Snowden, Representations of categories of $G$-maps, available at arXiv:1410.6054v3.
Lionel Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994. MR 1282727
Stefan Schwede, On the homotopy groups of symmetric spectra, Geom. Topol. 12 (2008), no. 3, 1313–1344. MR 2421129, DOI 10.2140/gt.2008.12.1313
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
Richard P. Stanley, Enumerative combinatorics. Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR 2868112
Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
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
John D. Wiltshire-Gordon, Uniformly presented vector spaces, available at arXiv:1406.0786v1.
Christopher F. Woodcock and Habib Sharif, On the transcendence of certain series, J. Algebra 121 (1989), no. 2, 364–369. MR 992771, DOI 10.1016/0021-8693(89)90072-0