Revised : 2018-01-21

Accepted : 2018-03-24

Published online : 2018-06-28

DOI : https://doi.org/10.5802/alco.18

Classification: 20C30, 05E15

Keywords: symmetric groups, double cover groups, characters, hook character, spin characters, Kronecker products, Saxl conjecture, unimodal sequences

@article{ALCO_2018__1_3_353_0, author = {Bessenrodt, Christine}, title = {Critical classes, Kronecker products of spin characters, and the Saxl conjecture}, journal = {Algebraic Combinatorics}, publisher = {MathOA foundation}, volume = {1}, number = {3}, year = {2018}, pages = {353-369}, doi = {10.5802/alco.18}, zbl = {06897705}, mrnumber = {3856528}, language = {en}, url = {https://alco.centre-mersenne.org/item/ALCO_2018__1_3_353_0} }

Critical classes, Kronecker products of spin characters, and the Saxl conjecture. Algebraic Combinatorics, Volume 1 (2018) no. 3, pp. 353-369. doi : 10.5802/alco.18. https://alco.centre-mersenne.org/item/ALCO_2018__1_3_353_0/

[1] Representations of the covering groups of the symmetric groups and their combinatorics, Sém. Lothar. Combin., Volume 33 (1994), B33a, 29 pages | MR 1411944 | Zbl 0855.05102

[2] On mixed products of complex characters of the double covers of the symmetric groups, Pac. J. Math., Volume 199 (2001) no. 2, pp. 257-268 | Article | MR 1847134 | Zbl 1056.20009

[3] On the Durfee size of Kronecker products of characters of the symmetric group and its double covers, J. Algebra, Volume 280 (2004) no. 1, pp. 132-144 | Article | MR 2081924 | Zbl 1073.20004

[4] On Kronecker products of complex representations of the symmetric and alternating groups, Pac. J. Math., Volume 190 (1999) no. 2, pp. 201-223 | Article | MR 1722888 | Zbl 1009.20013

[5] On Kronecker products of spin characters of the double covers of the symmetric groups, Pac. J. Math., Volume 198 (2001) no. 2, pp. 295-305 | Article | MR 1835510 | Zbl 1057.20006

[6] Spin representations and powers of 2, Algebr. Represent. Theory, Volume 3 (2000) no. 3, pp. 289-300 | Article | MR 1783803 | Zbl 1037.20011

[7] Spin representations, power of 2 and the Glaisher map, Algebr. Represent. Theory, Volume 8 (2005) no. 1, pp. 1-10 | Article | MR 2136918 | Zbl 1086.20008

[8] Kronecker coefficients for one hook shape, Sém. Lothar. Combin., Volume 77 (2017), B77c, 40 pages | MR 3650220 | Zbl 1361.05139

[9] On the Kronecker product of ${S}_{n}$ characters, J. Algebra, Volume 154 (1993) no. 1, pp. 125-140 | Article | MR 1201916 | Zbl 0848.20006

[10] Characters of finite groups, W. A. Benjamin (1967), viii+186 pages | MR 0219636 | Zbl 0166.29002

[11] A decomposition rule for certain tensor product representations of the symmetric groups, J. Algebra, Volume 434 (2015), pp. 46-64 | Article | MR 3342384 | Zbl 1310.05215

[12] Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type, Proc. Lond. Math. Soc., Volume 106 (2013) no. 4, pp. 908-930 | Article | MR 3056296 | Zbl 1372.20017

[13] Projective representations of the symmetric groups, Clarendon Press, Oxford Mathematical Monographs (1992), xiv+304 pages ($Q$-functions and shifted tableaux, Oxford Science Publications) | MR 1205350 | Zbl 0777.20005

[14] Lie algebraic proofs of some theorems on partitions, Number theory and algebra, Academic Press (1977), pp. 135-155 | MR 0491213 | Zbl 0372.10010

[15] The Saxl conjecture and the dominance order, Discrete Math., Volume 338 (2015) no. 11, pp. 1970-1975 | Article | MR 3357782 | Zbl 1314.05223

[16] The representation theory of the symmetric groups, Springer, Lecture Notes in Math., Volume 682 (1978), v+156 pages | MR 513828 | Zbl 0393.20009

[17] The representation theory of the symmetric group, Addison-Wesley Publishing Co., Encyclopedia of Mathematics and its Applications, Volume 16 (1981), xxviii+510 pages (with a foreword by P. M. Cohn and an introduction by Gilbert de B. Robinson) | MR 644144 | Zbl 0491.20010

[18] The projective characters of the symmetric groups that remain irreducible on subgroups, J. Algebra, Volume 138 (1991) no. 2, pp. 440-478 | Article | MR 1102819 | Zbl 0792.20012

[19] A simplified Kronecker rule for one hook shape, Proc. Am. Math. Soc., Volume 145 (2017) no. 9, pp. 3657-3664 | Article | MR 3665021 | Zbl 1365.05296

[20] The spin characters of the symmetric groups, Q. J. Math., Oxf. II Ser., Volume 13 (1962), pp. 241-246 | Article | MR 0144984 | Zbl 0112.02303

[21] The spin representation of the symmetric group, Canad. J. Math., Volume 17 (1965), pp. 543-549 | Article | MR 0175964 | Zbl 0135.05602

[22] On the unimodality of some partition polynomials, Eur. J. Comb., Volume 3 (1982) no. 1, pp. 69-84 | Article | MR 656013 | Zbl 0482.10015

[23] Bounds on certain classes of Kronecker and $q$-binomial coefficients, J. Comb. Theory, Ser. A, Volume 147 (2017), pp. 1-17 | Article | MR 3589885 | Zbl 1352.05011

[24] On the complexity of computing Kronecker coefficients, Comput. Complexity, Volume 26 (2017) no. 1, pp. 1-36 | Article | MR 3620776 | Zbl 1367.05012

[25] Kronecker products, characters, partitions, and the tensor square conjectures, Adv. Math., Volume 288 (2016), pp. 702-731 | Article | MR 3436397 | Zbl 1328.05199

[26] Lie superalgebras and some characters of ${S}_{n}$, Isr. J. Math., Volume 195 (2013) no. 1, pp. 31-35 | Article | MR 3101241 | Zbl 1279.20017

[27] Formulas for the expansion of the Kronecker products ${S}_{(m,n)}\otimes {S}_{({1}^{p-r},r)}$ and ${S}_{\left({1}^{k}{2}^{l}\right)}\otimes {S}_{({1}^{p-r},r)}$, Discrete Math., Volume 99 (1992) no. 1-3, pp. 265-287 | Article | MR 1158791 | Zbl 0755.05094

[28] The Kronecker product of Schur functions indexed by two-row shapes or hook shapes, J. Algebr. Comb., Volume 14 (2001) no. 2, pp. 153-173 | Article | MR 1867232 | Zbl 0999.05098

[29] The complex characters of the symmetric groups that remain irreducible in subgroups, J. Algebra, Volume 111 (1987) no. 1, pp. 210-219 | Article | MR 913206 | Zbl 0633.20008

[30] Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., Volume 139 (1911), pp. 155-250 | Article | MR 1580818 | Zbl 42.0154.02

[31] Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986), New York Academy of Sciences (Annals of the New York Academy of Sciences) Volume 576 (1989), pp. 500-535 | Article | MR 1110850 | Zbl 0792.05008

[32] Shifted tableaux and the projective representations of symmetric groups, Adv. Math., Volume 74 (1989) no. 1, pp. 87-134 | Article | MR 991411 | Zbl 0677.20012

[33] A note on skew characters of symmetric groups, Isr. J. Math., Volume 221 (2017) no. 1, pp. 435-443 | Article | MR 3705859 | Zbl 06808577

[34] GAP — Groups, Algorithms, and Programming, Version 4.7.4 (2014) (http://www.gap-system.org )