Defect $2$ spin blocks of symmetric groups and canonical basis coefficients
HTML articles powered by AMS MathViewer
- by Matthew Fayers PDF
- Represent. Theory 26 (2022), 134-178 Request permission
Abstract:
This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect $2$, analogous to Richards’s formula for defect $2$ blocks of symmetric groups.
By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding “$q$-decomposition numbers”, i.e. the canonical basis coefficients in the level-$1$ $q$-deformed Fock space of type $A^{(2)}_{2n}$; a special case of a conjecture of Leclerc and Thibon asserts that these coefficients yield the spin decomposition numbers in characteristic $2n+1$. Along the way, we prove some general results on $q$-decomposition numbers. This paper represents the first substantial progress on canonical bases in type $A^{(2)}_{2n}$.
References
- Christine Bessenrodt, Alun O. Morris, and Jørn B. Olsson, Decomposition matrices for spin characters of symmetric groups at characteristic $3$, J. Algebra 164 (1994), no. 1, 146–172. MR 1268331, DOI 10.1006/jabr.1994.1058
- 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, 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
- The GAP Group, GAP – Groups, Algorithms, and Programming, version 4.10.1, 2019, http://www.gap-system.org.
- P. N. Hoffman and J. F. Humphreys, Projective representations of the symmetric groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992. $Q$-functions and shifted tableaux; Oxford Science Publications. MR 1205350
- Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971, DOI 10.1090/gsm/042
- 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
- Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
- Seok-Jin Kang and Jae-Hoon Kwon, Fock space representations of quantum affine algebras and generalized Lascoux-Leclerc-Thibon algorithm, J. Korean Math. Soc. 45 (2008), no. 4, 1135–1202. MR 2422732, DOI 10.4134/JKMS.2008.45.4.1135
- M. Kashiwara, T. Miwa, J.-U. H. Petersen, and C. M. Yung, Perfect crystals and $q$-deformed Fock spaces, Selecta Math. (N.S.) 2 (1996), no. 3, 415–499. MR 1422203, DOI 10.1007/BF01587950
- Radha Kessar, Blocks and source algebras for the double covers of the symmetric and alternating groups, J. Algebra 186 (1996), no. 3, 872–933. MR 1424598, DOI 10.1006/jabr.1996.0400
- Radha Kessar and Mary Schaps, Crossover Morita equivalences for blocks of the covering groups of the symmetric and alternating groups, J. Group Theory 9 (2006), no. 6, 715–730. MR 2272713, DOI 10.1515/JGT.2006.046
- Ruthi Leabovich and Mary Schaps, Morita equivalences of spin blocks of symmetric and alternating groups, Rocky Mountain J. Math. 47 (2017), no. 3, 863–904. MR 3682153, DOI 10.1216/RMJ-2017-47-3-863
- Bernard Leclerc and Jean-Yves Thibon, $q$-deformed Fock spaces and modular representations of spin symmetric groups, J. Phys. A 30 (1997), no. 17, 6163–6176. MR 1482704, DOI 10.1088/0305-4470/30/17/023
- L. Maas, Modular spin characters of symmetric groups, Ph.D. thesis, Universität Duisburg–Essen, 2011.
- 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
- A. O. Morris and A. K. Yaseen, Decomposition matrices for spin characters of symmetric groups, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), no. 1-2, 145–164. MR 931015, DOI 10.1017/S0308210500026597
- 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
- M. L. Nazarov, Young’s orthogonal form of irreducible projective representations of the symmetric group, J. London Math. Soc. (2) 42 (1990), no. 3, 437–451. MR 1087219, DOI 10.1112/jlms/s2-42.3.437
- Matthew J. Richards, Some decomposition numbers for Hecke algebras of general linear groups, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 383–402. MR 1357053, DOI 10.1017/S0305004100074296
- 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
- Joanna Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), no. 2, 441–455. MR 1127075, DOI 10.1016/0021-8693(91)90319-4
- Joanna Scopes, Symmetric group blocks of defect two, Quart. J. Math. Oxford Ser. (2) 46 (1995), no. 182, 201–234. MR 1333832, DOI 10.1093/qmath/46.2.201
- K. M. Tan, Parities of $v$-decomposition numbers and an application to symmetric group algebras, arXiv:math/0606460, 2006.
- D. Yates, A further generalisation of bar-core partitions, arXiv:2107.04352, 2021.
Additional Information
- Matthew Fayers
- Affiliation: Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
- MR Author ID: 680587
- Email: m.fayers@qmul.ac.uk
- Received by editor(s): August 7, 2021
- Received by editor(s) in revised form: September 15, 2021
- Published electronically: March 17, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory 26 (2022), 134-178
- MSC (2020): Primary 20C30, 20C25, 17B37, 05E10
- DOI: https://doi.org/10.1090/ert/600
- MathSciNet review: 4396039