## Irreducible restrictions of representations of alternating groups in small characteristics: Reduction theorems

HTML articles powered by AMS MathViewer

- by Alexander Kleshchev, Lucia Morotti and Pham Huu Tiep
- Represent. Theory
**24**(2020), 115-150 - DOI: https://doi.org/10.1090/ert/538
- Published electronically: February 20, 2020
- PDF | Request permission

## Abstract:

We study irreducible restrictions from modules over alternating groups to proper subgroups, and prove reduction results which substantially restrict the classes of subgroups and modules for which this is possible. This problem had been solved when the characteristic of the ground field is greater than $3$, but the small characteristics cases require a substantially more delicate analysis and new ideas. This work fits into the Aschbacher-Scott program on maximal subgroups of finite classical groups.## References

- M. Aschbacher,
*On the maximal subgroups of the finite classical groups*, Invent. Math.**76**(1984), no. 3, 469–514. MR**746539**, DOI 10.1007/BF01388470 - Dave Benson,
*Spin modules for symmetric groups*, J. London Math. Soc. (2)**38**(1988), no. 2, 250–262. MR**966297**, DOI 10.1112/jlms/s2-38.2.250 - C. Bessenrodt and J. B. Olsson,
*On residue symbols and the Mullineux conjecture*, J. Algebraic Combin.**7**(1998), no. 3, 227–251. MR**1616083**, DOI 10.1023/A:1008618621557 - C. Bessenrodt and J. B. Olsson,
*Branching of modular representations of the alternating groups*, J. Algebra**209**(1998), no. 1, 143–174. MR**1652118**, DOI 10.1006/jabr.1998.7505 - C. Bessenrodt and J. B. Olsson,
*Residue symbols and Jantzen-Seitz partitions*, J. Combin. Theory Ser. A**81**(1998), no. 2, 201–230. MR**1603889**, DOI 10.1006/jcta.1997.2838 - John N. Bray, Derek F. Holt, and Colva M. Roney-Dougal,
*The maximal subgroups of the low-dimensional finite classical groups*, London Mathematical Society Lecture Note Series, vol. 407, Cambridge University Press, Cambridge, 2013. With a foreword by Martin Liebeck. MR**3098485**, DOI 10.1017/CBO9781139192576 - Jonathan Brundan and Alexander S. Kleshchev,
*Representations of the symmetric group which are irreducible over subgroups*, J. Reine Angew. Math.**530**(2001), 145–190. MR**1807270**, DOI 10.1515/crll.2001.002 - Peter J. Cameron, Peter M. Neumann, and Jan Saxl,
*An interchange property in finite permutation groups*, Bull. London Math. Soc.**11**(1979), no. 2, 161–169. MR**541970**, DOI 10.1112/blms/11.2.161 - Ben Ford and Alexander S. Kleshchev,
*A proof of the Mullineux conjecture*, Math. Z.**226**(1997), no. 2, 267–308. MR**1477629**, DOI 10.1007/PL00004340 - The GAP Group,
*GAP - Groups, Algorithms, and Programming*, Version 4.4; 2004, http://www.gap-system.org. - G. D. James,
*The Representation Theory of the Symmetric Groups*, Lecture Notes in Mathematics, vol.**682**, Springer, NewYork/Heidelberg/Berlin, 1978. - Jens C. Jantzen and Gary M. Seitz,
*On the representation theory of the symmetric groups*, Proc. London Math. Soc. (3)**65**(1992), no. 3, 475–504. MR**1182100**, DOI 10.1112/plms/s3-65.3.475 - Peter Kleidman and Martin Liebeck,
*The subgroup structure of the finite classical groups*, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR**1057341**, DOI 10.1017/CBO9780511629235 - Alexander S. Kleshchev,
*On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups. I*, Proc. London Math. Soc. (3)**69**(1994), no. 3, 515–540. MR**1289862**, DOI 10.1112/plms/s3-69.3.515 - Alexander Kleshchev,
*On decomposition numbers and branching coefficients for symmetric and special linear groups*, Proc. London Math. Soc. (3)**75**(1997), no. 3, 497–558. MR**1466660**, DOI 10.1112/S0024611597000427 - Alexander Kleshchev,
*Linear and projective representations of symmetric groups*, Cambridge Tracts in Mathematics, vol. 163, Cambridge University Press, Cambridge, 2005. MR**2165457**, DOI 10.1017/CBO9780511542800 - Alexander Kleshchev, Lucia Morotti, and Pham Huu Tiep,
*Irreducible restrictions of representations of symmetric groups in small characteristics: reduction theorems*, Math. Z.**293**(2019), no. 1-2, 677–723. MR**4002296**, DOI 10.1007/s00209-018-2203-1 - A.S. Kleshchev, L. Morotti and P.H. Tiep, Irreducible restrictions of representations of symmetric and alternating groups in small characteristics, arXiv:1903.09854.
- A. S. Kleshchev and J. K. Sheth,
*Representations of the symmetric group are reducible over simply transitive subgroups*, Math. Z.**235**(2000), no. 1, 99–109. MR**1785073**, DOI 10.1007/s002090000125 - Alexander S. Kleshchev and Jagat Sheth,
*Representations of the alternating group which are irreducible over subgroups*, Proc. London Math. Soc. (3)**84**(2002), no. 1, 194–212. MR**1863400**, DOI 10.1112/S002461150101320X - A. S. Kleshchev and A. E. Zalesski,
*Minimal polynomials of elements of order $p$ in $p$-modular projective representations of alternating groups*, Proc. Amer. Math. Soc.**132**(2004), no. 6, 1605–1612. MR**2051120**, DOI 10.1090/S0002-9939-03-07242-3 - Kay Magaard,
*Some remarks on maximal subgroups of finite classical groups*, Finite simple groups: thirty years of the atlas and beyond, Contemp. Math., vol. 694, Amer. Math. Soc., Providence, RI, 2017, pp. 123–137. MR**3682594**, DOI 10.1090/conm/694 - L. Morotti,
*Irreducible tensor products for alternating groups in characteristic 5*, Algebr. Represent. Theory (to appear). - G. Mullineux,
*Bijections of $p$-regular partitions and $p$-modular irreducibles of the symmetric groups*, J. London Math. Soc. (2)**20**(1979), no. 1, 60–66. MR**545202**, DOI 10.1112/jlms/s2-20.1.60 - Jan Saxl,
*The complex characters of the symmetric groups that remain irreducible in subgroups*, J. Algebra**111**(1987), no. 1, 210–219. MR**913206**, DOI 10.1016/0021-8693(87)90251-1 - Leonard L. Scott,
*Representations in characteristic $p$*, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 319–331. MR**604599** - David B. Wales,
*Some projective representations of $S_{n}$*, J. Algebra**61**(1979), no. 1, 37–57. MR**554850**, DOI 10.1016/0021-8693(79)90304-1

## Bibliographic Information

**Alexander Kleshchev**- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
- MR Author ID: 268538
- Email: klesh@uoregon.edu
**Lucia Morotti**- Affiliation: Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, 30167 Hannover, Germany
- MR Author ID: 1037296
- Email: morotti@math.uni-hannover.de
**Pham Huu Tiep**- Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
- MR Author ID: 230310
- Email: tiep@math.rutgers.edu
- Received by editor(s): September 25, 2019
- Received by editor(s) in revised form: January 10, 2020
- Published electronically: February 20, 2020
- Additional Notes: The first author was supported by the NSF grant DMS-1700905 and the DFG Mercator program through the University of Stuttgart.

This work was also supported by the NSF grant DMS-1440140 and the Simons Foundation while all three authors were in residence at the MSRI during the Spring 2018 semester.

The second author was supported by the DFG grant MO 3377/1-1, and the DFG Mercator program through the University of Stuttgart.

The third author was supported by the NSF grants DMS-1839351 and DMS-1840702, and the Joshua Barlaz Chair in Mathematics. - © Copyright 2020 American Mathematical Society
- Journal: Represent. Theory
**24**(2020), 115-150 - MSC (2010): Primary 20C20, 20C30, 20E28
- DOI: https://doi.org/10.1090/ert/538
- MathSciNet review: 4066477