Irreducible tensor products of representations of symmetric and related groups

• Applicant: Dr. Lucia Morotti
• Subject Area: Mathematics
• Term: from 2017 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 392927392

• Final Report Year: 2021

Final Report Abstract

The objective of the project was to complete the classification of irreducible tensor products of representations of alternating groups as well as of covering groups of symmetric and alternating groups. The last open parts in the corresponding classification for symmetric groups had been completed by the applicant shortly before applying for the first funding period. For alternating groups only the cases of characteristics 2, 3 and 5 were still open. For covering groups on the other hand arbitrary characteristic still had to be considered, with only some reduction results as well as results in characteristic 0 for covering groups of symmetric groups being known. In the time between the application for the first funding period of the project and the end of the second funding period the question has been completely solved. Apart for few exceptional cases, the classifications only consist of few easily described families of tensor products. Joint with Alexander Kleshchev and Pham Huu Tiep the closely related problem of classifying irreducible restrictions of representations of symmetric and alternating groups to arbitrary subgroups has also been essentially completed, by considering the (mostly) still open cases of characteristics 2 and 3. Both these classification questions have applications to the Aschbacher-Scott program on maximal subgroups of finite classical groups. As side results to these problems, some results on decomposition numbers as well as lower bounds on dimensions of irreducible representations have also been obtained. Other results that were obtained during the project are joint results with Haralampos Geranios and Alexander Kleshchev on self extensions of irreducible representations of symmetric groups as well as results considering character values in characteristic 0, which have been obtained partly joint with Hung P. Tong-Viet.

Publications

• Irreducible restrictions of representations of symmetric groups in small characteristics: reduction theorems, Math. Z. 293 (2019), 677–723
  A. Kleshchev, L. Morotti, P.H. Tiep
  (See online at https://doi.org/10.1007/s00209-018-2203-1)
• Composition factors of 2-parts spin representations of symmetric groups, Algebr. Comb. 3 (2020), 1283–1291
  L. Morotti
  (See online at https://doi.org/10.5802/alco.137)
• Irreducible restrictions of representations of alternating groups in small characteristics: reduction theorems, Represent. Theory 24 (2020), 115–150
  A. Kleshchev, L. Morotti, P.H. Tiep
  (See online at https://doi.org/10.1090/ert/538)
• Irreducible restrictions of representations of symmetric and alternating groups in small characteristics, Adv. Math. 369 (2020), 107184
  A. Kleshchev, L. Morotti, P.H. Tiep
  (See online at https://doi.org/10.1016/j.aim.2020.107184)
• Irreducible tensor products for alternating groups in characteristics 2 and 3, J. Pure Appl. Algebra 224 (2020), 106426
  L. Morotti
  (See online at https://doi.org/10.1016/j.jpaa.2020.106426)
• Irreducible tensor products for alternating groups in characteristic 5, Algebr. Represent. Theory 24 (2021), 203–229
  L. Morotti
  (See online at https://doi.org/10.1007/s10468-019-09941-0)
• Irreducible tensor products of representations of covering groups of symmetric and alternating groups, Represent. Theory, 25 (2021), 543–593
  L. Morotti
  (See online at https://doi.org/10.1090/ert/576)