Abstract
In this paper we study the asymptotic behavior of the number of summands in tensor products of finite dimensional representations of affine (semi)group (super)schemes and related objects.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Availability of data and materials
Not applicable.
References
Andersen, H.H.: The Steinberg linkage class for a reductive algebraic group. Ark. Mat. 56(2), 229–241 (2018) . https://doi.org/10.4310/ARKIV.2018.v56.n2.a2. arXiv:1706.00590
Andersen, H.H., Stroppel, C., Tubbenhauer, D.: Cellular structures using \(\rm U_q\)-tilting modules. Pacific J. Math. 292(1), 21–59 (2018). https://doi.org/10.2140/pjm.2018.292.21. arXiv:1503.00224
Andersen, H.H., Stroppel, C., Tubbenhauer, D.: Semisimplicity of Hecke and (walled) Brauer algebras. J. Aust. Math. Soc. 103(1), 1–44 (2017). https://doi.org/10.1017/S1446788716000392. arXiv:1507.07676
Benson, D.J., Doty, S.: Schur-Weyl duality over finite fields. Arch. Math. (Basel) 93(5), 425–435 (2009). https://doi.org/10.1007/s00013-009-0066-8. arXiv:0805.1235
Benson, D., Symonds, P.: The non-projective part of the tensor powers of a module. J. Lond. Math. Soc. (2) 101(2), 828–856 (2020). https://doi.org/10.1112/jlms.12288, arXiv:1902.02895
Berele, A., Regev, A.: Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras. Adv. in Math. 64(2), 118–175 (1987). https://doi.org/10.1016/0001-8708(87)90007-7
Bichon, J.: The representation category of the quantum group of a non-degenerate bilinear form. Comm. Algebra 31(10), 4831–4851 (2003). https://doi.org/10.1081/AGB-120023135arXiv:math/0111114
Brundan, J., Kleshchev, A.: Modular representations of the supergroup \(Q(n)\). I. vol. 260, pp. 64–98 (2003). Special issue celebrating the 80th birthday of Robert Steinberg. https://doi.org/10.1016/S0021-8693(02)00620-8
Brundan, J., Kujawa, J.: A new proof of the Mullineux conjecture. J. Algebraic Combin. 18(1), 13–39 (2003). https://doi.org/10.1023/A:1025113308552. arXiv:math/0210108
Bryant, R.M., Kovács, L.G.: Tensor products of representations of finite groups. Bull. London Math. Soc. 4, 133–135 (1972). https://doi.org/10.1112/blms/4.2.133
Coulembier, K., Etingof, P., Kleshchev, A., Ostrik, V.: Super invariant theory in positive characteristic. Eur. J. Math. 9(4), 94 (2023). https://doi.org/10.1007/s40879-023-00688-z. arXiv:2211.11933
Coulembier, K., Etingof, P., Ostrik, V.: Ann. of Math. (2) 197, no.3, 1235–1279 (2023). With Appendix A by A. Kleshchev. https://doi.org/10.4007/annals.2023.197.3.5, arXiv:2107.02372
Deligne, P.: Catégories tannakiennes. In The Grothendieck Festschrift, Vol. II, volume 87 of Progr. Math. pp 111–195. Birkhäuser Boston, Boston, MA, (1990)
Deligne, P.: La catégorie des représentations du groupe symétrique \(S_t\), lorsque \(t\) n’est pas un entier naturel. In Algebraic groups and homogeneous spaces, vol. 19 of Tata Inst. Fund. Res. Stud. Math. pp. 209–273. Tata Inst. Fund. Res. Mumbai, (2007)
Donkin, S.: On tilting modules for algebraic groups. Math. Z. 212(1), 39–60 (1993). https://doi.org/10.1007/BF02571640
Eger, S.: Stirling’s approximation for central extended binomial coefficients. Amer. Math. Monthly 121(4), 344–349 (2014). https://doi.org/10.4169/amer.math.monthly.121.04.344. arXiv:1203.2122
Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories, vol. 205 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, (2015). https://doi.org/10.1090/surv/205
Fulton, W.: Young tableaux, vol. 35 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, (1997). With applications to representation theory and geometry
Fulton, W., Harris, J.: Representation theory, vol. 129 of Graduate Texts in Mathematics. Springer-Verlag, New York,: A first course. Readings in Mathematics (1991). https://doi.org/10.1007/978-1-4612-0979-9
James, G., Kerber, A.: The representation theory of the symmetric group, vol. 16 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Co., Reading, Mass., With a foreword by P. M. Cohn, With an introduction by Gilbert de B. Robinson (1981)
James, G., Mathas, A.: A \(q\)-analogue of the Jantzen–Schaper theorem. Proc. London Math. Soc. (3) 74(2), 241–274 (1997). https://doi.org/10.1112/S0024611597000099
Jantzen, J.C.: Representations of algebraic groups, vol. 107 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, (2003)
Jensen, L.T.: Correction of the Lusztig–Williamson billiards conjecture. (2021). arXiv:2105.04665
Kolb, S.: Quantum symmetric Kac-Moody pairs. Adv. Math. 267, 395–469 (2014). https://doi.org/10.1016/j.aim.2014.08.010. arXiv:1207.6036
Khovanov, M., Sitaraman, M., Tubbenhauer, D.: Monoidal categories, representation gap and cryptography. To appear in Trans. Amer. Math. Soc. (2022). arXiv:2201.01805
Letzter, G.: Symmetric pairs for quantized enveloping algebras. J. Algebra 220(2), 729–767 (1999). https://doi.org/10.1006/jabr.1999.8015
Lusztig, G.: Quantum groups at roots of \(1\). Geom. Dedicata. 35(1–3), 89–113 (1990). https://doi.org/10.1007/BF00147341
Lusztig, G., Williamson, G.: Billiards and tilting characters for \(\rm SL_3\). SIGMA Symmetry Integrability Geom. Methods Appl. 14(015), 22 (2018). https://doi.org/10.3842/SIGMA.2018.015arXiv:1703.05898
Masuoka, A.: Harish-Chandra pairs for algebraic affine supergroup schemes over an arbitrary field. Transform. Groups 17(4), 1085–1121 (2012). https://doi.org/10.1007/s00031-012-9203-8. arXiv:1111.2387
Mathas, A.: Iwahori–Hecke algebras and Schur algebras of the symmetric group, vol. 15 of University Lecture Series. American Mathematical Society, Providence, RI, (1999). https://doi.org/10.1090/ulect/015
Milne, J.S.: Algebraic groups, vol. 170 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, The theory of group schemes of finite type over a field (2017). https://doi-org.ezproxy.library.sydney.edu.au/10.1017/9781316711736, https://doi.org/10.1017/9781316711736
Musson, I.M.: Lie superalgebras and enveloping algebras, vol. 131 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, (2012). https://doi.org/10.1090/gsm/131
Noumi, M., Sugitani, T.: Quantum symmetric spaces and related \(q\)-orthogonal polynomials. In Group theoretical methods in physics (Toyonaka, 1994), pp. 28–40. World Sci. Publ., River Edge, NJ, (1995)
Ostrik, V.: On symmetric fusion categories in positive characteristic. Selecta Math. (N.S.), 26(3):Paper No. 36, 19, (2020). https://doi.org/10.1007/s00029-020-00567-5, arXiv:1503.01492
Postnova, O., Reshetikhin, N.: On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras. Lett. Math. Phys. 110(1), 147–178 (2020). https://doi.org/10.1007/s11005-019-01217-4. arXiv:1812.11236
Rumer, G., Teller, E., Weyl, H.: Eine für die Valenztheorie geeignete Basis der binären Vektorinvarianten. Nachrichten von der Ges. der Wiss. Zu Göttingen. Math.-Phys. Klasse, pp. 498–504, In German (1932)
Sawin, S.F.: Quantum groups at roots of unity and modularity. J. Knot Theory Ramifications 15(10), 1245–1277 (2006). https://doi.org/10.1142/S0218216506005160, arXiv:math/0308281
Soergel, W.: Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules. Represent. Theory 1, 83–114 (1997). https://doi.org/10.1090/S1088-4165-97-00021-6
Soergel, W.: Character formulas for tilting modules over Kac-Moody algebras. Represent. Theory 2, 432–448 (1998). https://doi.org/10.1090/S1088-4165-98-00057-0
Spencer, R.A.: The modular Temperley–Lieb algebra. Rocky Mountain J. Math. 53(1), 177–208 (2023). https://doi.org/10.1216/rmj.2023.53.177. arXiv:2011.01328
Steinberg, B.: Representation theory of finite monoids. Universitext. Springer, Cham, (2016). https://doi.org/10.1007/978-3-319-43932-7
Stroppel, C.:Untersuchungen zu den parabolischen Kazhdan–Lusztig-Polynomen für affine Weyl-Gruppen. Diploma Thesis (1997), 74 pages (German) (1997). http://www.math.uni-bonn.de/ag/stroppel/arbeit_Stroppel.pdf
Sutton, L., Tubbenhauer, D., Wedrich, P., Zhu, J.: Sl2 tilting modules in the mixed case. Selecta Math. (N.S.) 29(3), 39 (2023). https://doi.org/10.1007/s00029-023-00835-0, arXiv:2105.07724
Tubbenhauer, D., Wedrich, P.: Quivers for \(\text{SL}_{2}\) tilting modules. Represent. Theory, 25, 440–480 (2021). https://doi.org/10.1090/ert/569, arXiv:1907.11560
Acknowledgements
Want to thank Pavel Etingof for comments on a draft of this paper, in particular for Remark 6.2 which is Pavel’s observation, Andrew Mathas for help with the literature on symmetric groups and Schur algebras, Volodymyr Mazorchuk for email exchanges about monoids, and Jonathan Gruber and Arun Ram for discussions about growth rates. We also thank the referee for very valuable comments and remarks. We also thank the MFO workshop 2235 “Character Theory and Categorification” for bringing us together in Oberwolfach in August/September 2022 – this project started during this fantastic workshop. V.O. and D.T. were supported by their depressions.
Funding
Open Access funding enabled and organized by CAUL and its Member Institutions. K.C. was partly supported by ARC grant DP200100712. D.T. was supported, in part, by the Australian Research Council.
Author information
Authors and Affiliations
Contributions
All authors contributed equally with respect to every section of the paper.
Corresponding author
Ethics declarations
Ethical Approval
Not applicable.
Competing interests
No financial or personal competing interest.
Additional information
Presented by: Alistair Savage
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Coulembier, K., Ostrik, V. & Tubbenhauer, D. Growth Rates of the Number of Indecomposable Summands in Tensor Powers. Algebr Represent Theor 27, 1033–1062 (2024). https://doi.org/10.1007/s10468-023-10245-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-023-10245-7
Keywords
- Tensor products
- Asymptotic behavior
- Affine group schemes
- Affine semigroup schemes
- Semigroups
- Supergroups
- Hopf algebras
- (Symmetric) monoidal categories