Stuttering blocks of Ariki–Koike algebras
Algebraic Combinatorics, Tome 2 (2019) no. 1, pp. 75-118.

We study a shift action defined on multipartitions and on residue multisets of their Young diagrams. We prove that the minimal orbit cardinality among all multipartitions associated with a given multiset depends only on the orbit cardinality of the multiset. Using abaci, this problem reduces to a convex optimisation problem over the integers with linear constraints. We solve it by proving an existence theorem for binary matrices with prescribed row, column and block sums. Finally, we give some applications to the representation theory of the Hecke algebra of the complex reflection group G(r,p,n).

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/alco.40
Classification : 20C08
Mots clés : Ariki–Koike algebras, multipartitions, residues, abacus, Hecke algebras
Rostam, Salim 1

1 Laboratoire de Mathématiques de Versailles UVSQ CNRS Université Paris-Saclay 78035 Versailles (France)
@article{ALCO_2019__2_1_75_0,
     author = {Rostam, Salim},
     title = {Stuttering blocks of {Ariki{\textendash}Koike} algebras},
     journal = {Algebraic Combinatorics},
     pages = {75--118},
     publisher = {MathOA foundation},
     volume = {2},
     number = {1},
     year = {2019},
     doi = {10.5802/alco.40},
     mrnumber = {3912169},
     zbl = {1425.20006},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/alco.40/}
}
TY  - JOUR
AU  - Rostam, Salim
TI  - Stuttering blocks of Ariki–Koike algebras
JO  - Algebraic Combinatorics
PY  - 2019
SP  - 75
EP  - 118
VL  - 2
IS  - 1
PB  - MathOA foundation
UR  - http://www.numdam.org/articles/10.5802/alco.40/
DO  - 10.5802/alco.40
LA  - en
ID  - ALCO_2019__2_1_75_0
ER  - 
%0 Journal Article
%A Rostam, Salim
%T Stuttering blocks of Ariki–Koike algebras
%J Algebraic Combinatorics
%D 2019
%P 75-118
%V 2
%N 1
%I MathOA foundation
%U http://www.numdam.org/articles/10.5802/alco.40/
%R 10.5802/alco.40
%G en
%F ALCO_2019__2_1_75_0
Rostam, Salim. Stuttering blocks of Ariki–Koike algebras. Algebraic Combinatorics, Tome 2 (2019) no. 1, pp. 75-118. doi : 10.5802/alco.40. http://www.numdam.org/articles/10.5802/alco.40/

[1] Ariki, Susumu On the semi-simplicity of the Hecke algebra of (/r)𝔖 n , J. Algebra, Volume 169 (1994) no. 1, pp. 216-225 | DOI | MR | Zbl

[2] Ariki, Susumu Representation theory of a Hecke algebra of G(r,p,n), J. Algebra, Volume 177 (1995) no. 1, pp. 164-185 | DOI | MR | Zbl

[3] Ariki, Susumu On the classification of simple modules for cyclotomic Hecke algebras of type G(m,1,n) and Kleshchev multipartitions, Osaka J. Math., Volume 38 (2001) no. 4, pp. 827-837 | MR | Zbl

[4] Ariki, Susumu; Koike, Kazuhiko A Hecke algebra of (/r)𝔖 n and construction of its irreducible representations, Adv. Math., Volume 106 (1994) no. 2, pp. 216-243 | DOI | MR | Zbl

[5] Ariki, Susumu; Mathas, Andrew The number of simple modules of the Hecke algebras of type G(r,1,n), Math. Z., Volume 233 (2000) no. 3, pp. 601-623 | DOI | MR | Zbl

[6] Bowman, Chris The many graded cellular bases of Hecke algebras (2017) (https://arxiv.org/abs/1702.06579)

[7] Broué, Michel; Malle, Gunter; Rouquier, Raphaël Complex Reflection Groups, Braid Groups, Hecke Algebras, J. Reine Angew. Math., Volume 1998 (2006) no. 500, pp. 127-190 (Accessed 2017-02-13) | DOI | Zbl

[8] Brundan, Jonathan; Kleshchev, Alexander Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras, Invent. Math., Volume 178 (2009) no. 3, pp. 451-484 | DOI | MR | Zbl

[9] Chernyak, Zhanna A.; Chernyak, Arkady A. Matrices with prescribed row, column and block sums, Combinatorica, Volume 8 (1988) no. 2, pp. 177-184 | DOI | MR | Zbl

[10] Chlouveraki, Maria; Jacon, Nicolas Schur Elements for the Ariki–Koike Algebra and Applications, J. Algebr. Comb., Volume 35 (2012) no. 2, pp. 291-311 | DOI | MR | Zbl

[11] Dipper, Richard; James, Gordon; Mathas, Andrew Cyclotomic q-Schur Algebras, Math. Z., Volume 229 (1998) no. 3, pp. 385-416 | DOI | MR | Zbl

[12] Dipper, Richard; Mathas, Andrew Morita Equivalences of Ariki–Koike Algebras, Math. Z., Volume 240 (2002) no. 3, pp. 579-610 | DOI | MR | Zbl

[13] Fayers, Matthew Weights of multipartitions and representations of Ariki–Koike algebras, Adv. Math., Volume 206 (2006) no. 1, pp. 112-144 (an updated version of this paper is available from http://www.maths.qmul.ac.uk/~mf/) | DOI | MR | Zbl

[14] Gale, David A theorem on flows in networks, Pac. J. Math., Volume 7 (1957) no. 2, pp. 1073-1082 | DOI | MR | Zbl

[15] Garvan, Frank; Kim, Dongsu; Stanton, Dennis Cranks and t-cores, Invent. Math., Volume 101 (1990) no. 1, pp. 1-17 | DOI | MR | Zbl

[16] Genet, Gwenaëlle; Jacon, Nicolas Modular Representations of Cyclotomic Hecke Algebras of Type G(r,p,n), Int. Math. Res. Not. (2006), O93049, 18 pages | DOI | MR | Zbl

[17] Graham, John J.; Lehrer, Gustav I. Cellular algebras, Invent. Math., Volume 123 (1996) no. 1, pp. 1-34 | DOI | MR | Zbl

[18] Hu, Ju; Mathas, Andrew Graded Cellular Bases for the Cyclotomic Khovanov-Lauda-Rouquier Algebras of Type A, Adv. Math., Volume 225 (2010) no. 2, pp. 598-642 | DOI | MR | Zbl

[19] Hu, Ju; Mathas, Andrew Decomposition numbers for Hecke algebras of type G(r,p,n): the (ε,q)-separated case, Proc. Lond. Math. Soc., Volume 104 (2012) no. 5, pp. 865-926 | DOI | MR | Zbl

[20] James, Gordon Some combinatorial results involving Young diagrams, Math. Proc. Camb. Philos. Soc., Volume 83 (1978) no. 1, pp. 1-10 | DOI | MR | Zbl

[21] James, Gordon; Kerber, Adalbert The representation theory of the symmetric group, Encyclopedia of Mathematics and Its Applications, 16, Addison-Wesley Publishing, 1981 | MR | Zbl

[22] Lyle, Sinéad; Mathas, Andrew Blocks of Cyclotomic Hecke Algebras, Adv. Math., Volume 216 (2007) no. 2, pp. 854-878 | DOI | MR | Zbl

[23] Mathas, Andrew Iwahori–Hecke algebras and Schur algebras of the symmetric group, American Mathematical Society, 1999 | Zbl

[24] Merentes, Nelson; Nikodem, Kazimierz Remarks on strongly convex functions, Aequationes Math., Volume 80 (2010) no. 1-2, pp. 193-199 | DOI | MR | Zbl

[25] Olsson, Jørn B. Combinatorics and representations of finite groups, Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, 20, Universität Essen, 1993, ii+94 pages | MR | Zbl

[26] Rostam, Salim Cyclotomic quiver Hecke algebras and Hecke algebra of G(r,p,n) (2016) (https://arxiv.org/abs/1609.08908, to appear in Trans. Amer. Math. Soc.) | Zbl

[27] Rouquier, Raphaël 2-Kac–Moody algebras (2008) (https://arxiv.org/abs/0812.5023)

[28] Ryser, Herbert J. Combinatorial properties of matrices of zeroes and ones, Can. J. Math., Volume 9 (1957), pp. 371-377 | DOI | Zbl

[29] Wada, Kentaro Blocks of category 𝒪 for rational Cherednik algebras and of cyclotomic Hecke algebras of type G(r,p,n), Osaka J. Math., Volume 48 (2011) no. 4, pp. 895-912 | MR | Zbl

Cité par Sources :