Spin Characters' Decomposition Matrices of S27, S28modulo, p=13
Main Article Content
Abstract
In this study, when the field characteristic is 13, we calculate decomposition matrices for the spin characters and which are broken down into blocks, where the decomposition matrices are connected between irreducible spin characters and irreducible modular spin characters. The technique used in this study is -inducing, which produces projective characters for symmetric group by projecting 's character, and symmetric group by projecting 's character. We can find it by fixing all bar divisions, finding all irreducible spin characters for ( ), , and all irreducible modular spin characters for ( ), . In order to explore irreducible modular spin characteristics, general correlations and theorems will be discovered as a result of this research.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
licenseTerms
Publication Dates
Received
Accepted
Published Online First
References
Schur , J. Uber die Darstellung der symmetrischen und der alternierenden gruppe durch gebrochene lineare subtituttionen. J. Reine ang.Math. 2009, 155-250. https://doi.org/10.1515/crll.1911.139.155
Bessenrodt, C., Morris, A. O., and Olsson, J. B. Decomposition matrices for spin characters of symmetric groups at characteristic 3. Journal of. Algebra. 1994, 164(1), 146–172. https://doi.org/10.1006/jabr.1994.1058
Issacs, I. M. Character theory of finite groups. Academic press, INC, 1976.
Morris, O. A. and Yaseen, A. K. Decomposition matrices for spin characters of symmetric group. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 1988, 108, 145-164. https://doi.org/10.1017/S0308210500026597.
Morris, A. O. The spin representation of the symmetric group.Proc. Canadian Journal of Mathematics. 1962, 12, 55–76. https://doi.org/10.1112/plms/s3-12.1.55
Morris, A. O. The spin representation of the symmetric group. Canadian Journal of Mathematics 1965, 17, 543–549. https://doi.org/10.4153/CJM-1965-055-0.
Yaseen, A. K. Modular spin representations of the symmetric groups. Doctoral dissertation, The University of Wales: United Kingdom, 1987.
Humphreys, F.J. Blocks of the Projective representations of symmetric groups. London Mathematical Societ, 1986, 441-452. https://doi.org/10.1112/jlms/s2-33.3.441
Yaseen, A. K. The Brauer trees of the symmetric group S21 modulo p=13. Basrah Journal of Scienc, 2019, 1, 126-140. https://doi.org/10.29072/basjs.20190110.
Sharqi, S.M. Modular Spin Characters for Some Symmetric Groups. Master's thesis. Basrah University:Iraq, 2019.
Jassim, A. H , and Taban, S. A. Spin Characters' Decomposition Matrix, S24 modulo, p=7. Journal of Basrah Researches (Sciences) 2023, 49(1), 66–83. https://doi.org/10.56714/bjrs.49.1.7
Jassim, A. H , and Taban, S. A. Decomposition Matrix for the projective Characters S28, p=11, Journal of Kufa for Mathematics and Computer 2024, 11(1), 70-82.
http://dx.doi.org/10.31642/JoKMC/2018/110112
Yaseen, A. K.; Tahir, M. B. 13-brauer trees of the symmetric group S22. Appl. Math. Inf. Sci. 2020, 14, 327–334. https://doi.org/10.29072/basjs.20190110.
Fayers, M.; Morotti, L. On the irreducible spin representations of symmetric and alternating groups which remain irreducible in characteristic 3. Representation Theory of the American Mathematical Society 2023, 27, 778-814. https://doi.org/10.1090/ert/654.
Kleshchev, A.; Morotti, L.; Tiep, P.H. Irreducible restrictions of representations of symmetric and alternating groups in small characteristics. Advances in Mathematics 2020, 369, 1-66 https://doi.org/10.1016/j.aim.2020.107184.
Morotti, Lucia. Composition factors of 2-parts spin representations of symmetric groups, Algebraic Combinatorics 2020, 3(6), 1283-1291. https://doi.org/10.5802/alco.137.
Kazuya Aokage. Tensor square of the basic spin representations of Schur covering groups for the symmetric groups, Journal of Algebraic Combinatorics 2020, 54(1), 135-150, https://doi.org/10.1007/s10801-020-00972-1
Haggarty, R. J.; Humphreys, J. F. Projective Characters of Finite Groups. Proceedings of the London Mathematical Society 1978, 36(1), 176–192. https://doi.org/10.1112/plms/s3-36.1.176.
Brundan, J.; Kleshchev, A. S. Representations of the symmetric group which are irreducible over subgroups, Journal Für Die Reine Und Angewandte Math. 2001, 145- 190, 530. https://doi.org/10.1515/crll.2001.002 .
Morotti, L. Irreducible Tensor Products for Alternating Groups in Characteristic 5. Algebras and Representation Theory 2020, 24(1), 203–229. https://doi.org/10.1007/s10468-019-09941-0 .
Morotti, L. Irreducible Tensor Products Of Representations Of Covering Groups Of Symmetric And Alternating Groups, Journal of the American Mathematical Society 2021, 25, 543–593. https://doi.org/10.1090/ert/576.
Maas, L. A. Modular Spin Characters of Symmetric Groups. Doctoral dissertation, University at Duisburg Essen, 2011.
Puttaswamaiah, B. M., and Dixon, D.J. Modular representation of finite groups. New York. Academic Press, 1977.
James, D. G. The modular characters of the Mathieu groups. Journal of Algebra 1973, 27, 57-111. https://doi.org/10.1016/0021-8693(73)90165-8.
Brundan, J.; Kleshchev, A. S. Representations of the symmetric group which are irreducible over subgroups. J. reine angew.Math. 2001, 530, 145–190. https://doi.org/10.1515/crll.2001.002 .