TY - JOUR

T1 - On subsequences and certain elements which determine various cells in S_n

AU - Pallikaros, C. A.

AU - McDonough, Thomas

N1 - T.P.McDonough, C.A.Pallikaros,
On subsequences and certain elements which determine various cells in S_n,
Journal of Algebra,
2008, volume 319, issue 3,
pp. 1249-1263.

PY - 2008/2/1

Y1 - 2008/2/1

N2 - We study the relation between certain increasing and decreasing subsequences occurring in the row form of certain elements in the symmetric group, following Schensted [C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961) 179–191] and Greene [C. Greene, An extension of Schensted's theorem, Adv. Math. 14 (1974) 254–265], and the Kazhdan–Lusztig cells [D.A. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184] of the symmetric group to which they belong. We show that, in the two-sided cell corresponding to a partition λ, there is an explicitly defined element dλ, each of whose prefixes can be used to obtain a left cell by multiplying the cell containing the longest element of the parabolic subgroup associated with λ on the right. Furthermore, we show that the elements of these left cells are those which possess increasing and decreasing subsequences of certain types. The results in this paper lead to efficient algorithms for the explicit descriptions of many left cells inside a given two-sided cell, and the authors have implemented these algorithms in GAP.

AB - We study the relation between certain increasing and decreasing subsequences occurring in the row form of certain elements in the symmetric group, following Schensted [C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961) 179–191] and Greene [C. Greene, An extension of Schensted's theorem, Adv. Math. 14 (1974) 254–265], and the Kazhdan–Lusztig cells [D.A. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184] of the symmetric group to which they belong. We show that, in the two-sided cell corresponding to a partition λ, there is an explicitly defined element dλ, each of whose prefixes can be used to obtain a left cell by multiplying the cell containing the longest element of the parabolic subgroup associated with λ on the right. Furthermore, we show that the elements of these left cells are those which possess increasing and decreasing subsequences of certain types. The results in this paper lead to efficient algorithms for the explicit descriptions of many left cells inside a given two-sided cell, and the authors have implemented these algorithms in GAP.

KW - Symmetric groups

KW - Young tableaux

KW - Kazhdan–Lusztig cells

KW - Coxeter groups

U2 - 10.1016/j.jalgebra.2007.03.047

DO - 10.1016/j.jalgebra.2007.03.047

M3 - Article

SN - 1090-266X

VL - 319

SP - 1249

EP - 1263

JO - Journal of Algebra

JF - Journal of Algebra

IS - 3

ER -