Defect 2 spin blocks of symmetric groups and canonical basis coefficients

Fayers, Matthew

doi:10.1090/ert/600

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.