Article
| Title: | Modular separating invariants for the dihedral groups $D_{2p}$ (English) |
| Author: | Jia, Panpan |
| Author: | Nan, Jizhu |
| Author: | Ma, Yongsheng |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 4 |
| Year: | 2025 |
| Pages: | 1291-1306 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | Let $\mathbb {F}$ be an algebraically closed field of odd prime characteristic $p$. Using only transfers and norms, we describe a separating set for each indecomposable modular representation of the dihedral groups $D_{2p}$ over the field $\mathbb {F}$. Our construction is recursive and the size of every separating set depends only on the dimension of the representation. (English) |
| Keyword: | modular invariant |
| Keyword: | separating set |
| Keyword: | dihedral group |
| MSC: | 13A50 |
| DOI: | 10.21136/CMJ.2025.0076-25 |
| . | |
| Date available: | 2025-12-20T07:27:38Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153243 |
| . | |
| Reference: | [1] Campbell, H. E. A., Wehlau, D. L.: Modular Invariant Theory.Encyclopaedia of Mathematical Sciences 139. Invariant Theory and Algebraic Transformation Groups 8. Springer, Berlin (2011). Zbl 1216.14001, MR 2759466, 10.1007/978-3-642-17404-9 |
| Reference: | [2] Chen, H., Nan, J., Wang, Y.: Coinvariants and invariants of the dihedral group $D_{2p}$ in characteristic $p>2$.Linear Multilinear Algebra 66 (2018), 224-242. Zbl 1390.13019, MR 3750585, 10.1080/03081087.2017.1295431 |
| Reference: | [3] Chen, Y., Shank, R. J., Wehlau, D. L.: Modular invariants of finite gluing groups.J. Algebra 566 (2021), 405-434. Zbl 1459.13008, MR 4153231, 10.1016/j.jalgebra.2020.08.034 |
| Reference: | [4] Chen, Y., Zhang, X.: A class of quadratic matrix equations over finite fields.Algebra Colloq. 30 (2023), 169-180. Zbl 1515.15014, MR 4562922, 10.1142/s1005386723000147 |
| Reference: | [5] Derksen, H., Kemper, G.: Computational Invariant Theory.Encyclopaedia of Mathematical Sciences 130. Invariant Theory and Algebraic Transformation Groups 1. Springer, Berlin (2002). Zbl 1011.13003, MR 1918599, 10.1007/978-3-662-04958-7 |
| Reference: | [6] Draisma, J., Kemper, G., Wehlau, D.: Polarization of separating invariants.Can. J. Math. 60 (2008), 556-571. Zbl 1143.13008, MR 2414957, 10.4153/cjm-2008-027-2 |
| Reference: | [7] Dufresne, E.: Separating invariants and finite reflection groups.Adv. Math. 221 (2009), 1979-1989. Zbl 1173.13004, MR 2522833, 10.1016/j.aim.2009.03.013 |
| Reference: | [8] Dufresne, E., Elmer, J., Kohls, M.: The Cohen-Macaulay property of separating invariants of finite groups.Transform. Groups 14 (2009), 771-785. Zbl 1184.13019, MR 2577197, 10.1007/s00031-009-9072-y |
| Reference: | [9] Dufresne, E., Elmer, J., Sezer, M.: Separating invariants for arbitrary linear actions of the additive group.Manuscr. Math. 143 (2014), 207-219. Zbl 1294.13006, MR 3147449, 10.1007/s00229-013-0625-y |
| Reference: | [10] Dufresne, E., Jeffries, J.: Separating invariants and local cohomology.Adv. Math. 270 (2015), 565-581. Zbl 1318.13009, MR 3286543, 10.1016/j.aim.2014.11.003 |
| Reference: | [11] Elmer, J., Kohls, M.: Separating invariants for the basic $\Bbb{G}_a$-actions.Proc. Am. Math. Soc. 140 (2012), 135-146. Zbl 1241.13009, MR 2833525, 10.1090/s0002-9939-2011-11273-5 |
| Reference: | [12] Jia, P., Nan, J., Ma, Y.: Separating invariants for certain representations of the elementary Abelian $p$-groups of rank two.AIMS Math. 9 (2024), 25603-25618. MR 4795461, 10.3934/math.20241250 |
| Reference: | [13] Kemper, G.: Separating invariants.J. Symb. Comput. 44 (2009), 1212-1222. Zbl 1172.13001, MR 2532166, 10.1016/j.jsc.2008.02.012 |
| Reference: | [14] Kemper, G., Lopatin, A., Reimers, F.: Separating invariants over finite fields.J. Pure Appl. Algebra 226 (2022), Article ID 106904, 18 pages. Zbl 1478.13009, MR 4310052, 10.1016/j.jpaa.2021.106904 |
| Reference: | [15] Kohls, M., Kraft, H.: Degree bounds for separating invariants.Math. Res. Lett. 17 (2010), 1171-1182. Zbl 1230.13010, MR 2729640, 10.4310/mrl.2010.v17.n6.a15 |
| Reference: | [16] Kohls, M., Sezer, M.: Invariants of the dihedral group $D_{2p}$ in characteristic two.Math. Proc. Camb. Philos. Soc. 152 (2012), 1-7. Zbl 1234.13008, MR 2860414, 10.1017/s030500411100065x |
| Reference: | [17] Kohls, M., Sezer, M.: Separating invariants for the Klein four group and cyclic groups.Int. J. Math. 24 (2013), Article ID 1350046, 11 pages. Zbl 1311.13008, MR 3078070, 10.1142/s0129167x13500468 |
| Reference: | [18] Neusel, M. D., Sezer, M.: Separating invariants for modular $p$-groups and groups acting diagonally.Math. Res. Lett. 16 (2009), 1029-1036. Zbl 1194.13004, MR 2576691, 10.4310/mrl.2009.v16.n6.a11 |
| Reference: | [19] Reimers, F.: Separating invariants of finite groups.J. Algebra 507 (2018), 19-46. Zbl 1395.13002, MR 3807041, 10.1016/j.jalgebra.2018.03.022 |
| Reference: | [20] Reimers, F.: Separating invariants for two copies of the natural $S_n$-action.Commun. Algebra 48 (2020), 1584-1590. Zbl 1458.13007, MR 4079329, 10.1080/00927872.2019.1691575 |
| Reference: | [21] Schefler, B.: The separating Noether number of abelian groups of rank two.J. Comb. Theory, Ser. A 209 (2025), Article ID 105951, 15 pages. Zbl 07943111, MR 4792433, 10.1016/j.jcta.2024.105951 |
| Reference: | [22] Sezer, M.: Constructing modular separating invariants.J. Algebra 322 (2009), 4099-4104. Zbl 1205.13009, MR 2556140, 10.1016/j.jalgebra.2009.07.011 |
| Reference: | [23] Sezer, M.: Explicit separating invariants for cyclic $p$-groups.J. Comb. Theory, Ser. A 118 (2011), 681-689. Zbl 1267.13011, MR 2739512, 10.1016/j.jcta.2010.05.003 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
