On the irreducible spin representations of symmetric and alternating groups which remain irreducible in characteristic 3

Fayers, Matthew; Morotti, Lucia

doi:10.1090/ert/654

On the irreducible spin representations of symmetric and alternating groups which remain irreducible in characteristic $3$
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by Matthew Fayers and Lucia Morotti;

Represent. Theory 27 (2023), 778-814

DOI: https://doi.org/10.1090/ert/654

Published electronically: September 11, 2023

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Abstract:

For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$, or more generally, which representations remain homogeneous in characteristic $p$. In this paper we address this question when $G$ is a proper double cover of the symmetric or alternating group. We obtain a classification when $p=3$ except in the case of a certain family of partitions relating to spin RoCK blocks. Our techniques involve induction and restriction, degree calculations, decomposing projective characters and recent results of Kleshchev and Livesey on spin RoCK blocks.

References

Jonathan Brundan and Alexander Kleshchev, Hecke-Clifford superalgebras, crystals of type $A_{2l}^{(2)}$ and modular branching rules for $\hat S_n$, Represent. Theory 5 (2001), 317–403. MR 1870595, DOI 10.1090/S1088-4165-01-00123-6
Jonathan Brundan and Alexander Kleshchev, Projective representations of symmetric groups via Sergeev duality, Math. Z. 239 (2002), no. 1, 27–68. MR 1879328, DOI 10.1007/s002090100282
Jonathan Brundan and Alexander Kleshchev, Representation theory of symmetric groups and their double covers, Groups, combinatorics & geometry (Durham, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 31–53. MR 1994959, DOI 10.1142/9789812564481_{0}003
Jonathan Brundan and Alexander Kleshchev, James’ regularization theorem for double covers of symmetric groups, J. Algebra 306 (2006), no. 1, 128–137. MR 2271575, DOI 10.1016/j.jalgebra.2006.01.055
Joseph Chuang and Kai Meng Tan, Representations of wreath products of algebras, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 3, 395–411. MR 2018255, DOI 10.1017/S0305004103006984
Matthew Fayers, Reducible Specht modules, J. Algebra 280 (2004), no. 2, 500–504. MR 2089249, DOI 10.1016/j.jalgebra.2003.09.053
Matthew Fayers, Irreducible Specht modules for Hecke algebras of type $\textbf {A}$, Adv. Math. 193 (2005), no. 2, 438–452. MR 2137291, DOI 10.1016/j.aim.2004.06.001
Matthew Fayers, On the irreducible representations of the alternating group which remain irreducible in characteristic $p$, Represent. Theory 14 (2010), 601–626. MR 2685098, DOI 10.1090/S1088-4165-2010-00390-8
Matthew Fayers, The irreducible representations of the alternating group which remain irreducible in characteristic $p$, Trans. Amer. Math. Soc. 368 (2016), no. 8, 5807–5855. MR 3458400, DOI 10.1090/tran/6531
Matthew Fayers, Irreducible projective representations of the symmetric group which remain irreducible in characteristic 2, Proc. Lond. Math. Soc. (3) 116 (2018), no. 4, 878–928. MR 3789835, DOI 10.1112/plms.12087
Matthew Fayers, Irreducible projective representations of the alternating group which remain irreducible in characteristic 2, Adv. Math. 374 (2020), 107340, 62. MR 4131402, DOI 10.1016/j.aim.2020.107340
M. Fayers, A. Kleshchev, and L. Morotti, Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups, arXiv:2303.03304, 2023.
John F. Humphreys, Blocks of projective representations of the symmetric groups, J. London Math. Soc. (2) 33 (1986), no. 3, 441–452. MR 850960, DOI 10.1112/jlms/s2-33.3.441
G. D. James, On a conjecture of Carter concerning irreducible Specht modules, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 11–17. MR 463281, DOI 10.1017/S0305004100054232
Gordon James and Andrew Mathas, A $q$-analogue of the Jantzen-Schaper theorem, Proc. London Math. Soc. (3) 74 (1997), no. 2, 241–274. MR 1425323, DOI 10.1112/S0024611597000099
Gordon James and Andrew Mathas, The irreducible Specht modules in characteristic $2$, Bull. London Math. Soc. 31 (1999), no. 4, 457–462. MR 1687552, DOI 10.1112/S0024609399005822
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
A. Kleshchev and M. Livesey, RoCK blocks for double covers of symmetric groups and quiver Hecke superalgebras, arXiv:2201.06870, 2022.
Alexander Kleshchev and Vladimir Shchigolev, Modular branching rules for projective representations of symmetric groups and lowering operators for the supergroup $Q(n)$, Mem. Amer. Math. Soc. 220 (2012), no. 1034, xviii+123. MR 3014600, DOI 10.1090/S0065-9266-2012-00657-5
Alexander S. Kleshchev and Pham Huu Tiep, Representations of finite special linear groups in non-defining characteristic, Adv. Math. 220 (2009), no. 2, 478–504. MR 2466423, DOI 10.1016/j.aim.2008.09.011
Sinéad Lyle, Some reducible Specht modules, J. Algebra 269 (2003), no. 2, 536–543. MR 2015852, DOI 10.1016/S0021-8693(03)00537-4
L. A. Maas, Modular spin characters of symmetric groups, Ph.D. thesis, https://duepublico2.uni-due.de/receive/duepublico_mods_00027043, 2012.
A. O. Morris and A. K. Yaseen, Some combinatorial results involving shifted Young diagrams, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 23–31. MR 809494, DOI 10.1017/S030500410006388X
Jürgen Müller, Brauer trees for the Schur cover of the symmetric group, J. Algebra 266 (2003), no. 2, 427–445. MR 1995123, DOI 10.1016/S0021-8693(03)00342-9
J. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155–250 (German). MR 1580818, DOI 10.1515/crll.1911.139.155
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