# On double cosets with the trivial intersection property and Kazhdan-Lusztig cells in $S_n$

Document Type : Research Paper

Authors

1 Department of Mathematics, Aberystwyth University

2 Department of Mathematics and Statistics, University of Cyprus

Abstract

‎For a composition $\lambda$ of $n$ our aim is to obtain reduced forms‎ ‎for all the elements in the ‎$w_{J(\lambda)}$‎, ‎the longest element of the standard parabolic‎ ‎subgroup of $S_n$ corresponding to $\lambda$‎. ‎We investigate how far this is possible to achieve by looking at‎ ‎elements of the form $w_{J(\lambda)}d$‎, ‎where $d$ is a prefix of‎ ‎an element of minimum length in a $(W_{J(\lambda)},B)$ double coset‎ ‎with the trivial intersection property‎, ‎$B$ being a parabolic subgroup‎ ‎of $S_n$ whose type is `dual' to that of $W_{J(\lambda)}$‎.

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#### References

S. Ariki (2000). Robinson-Schensted corresp ondence and left cells. Adv. Stud. Pure Math.. 28, 1-20 A. Bjorner and F. Brenti (2005). Combinatorics of Coxeter groups. Graduate Texts in Mathematics, Springer-Verlag, New York. 231 T. Braden and R. MacPherson (2001). From moment graphs to intersection cohomology. Math. Ann.. 321, 533-551 C. W. Curtis and I. Reiner (1962). Representation theory of finite groups and associative algebras. Wiley, New York. C. W. Curtis and I. Reiner (1981/1987). Methods of representation theory, I & II. A Wiley-Interscience Publication. John Wiley Sons, Inc., New York. V. V. Deodhar (1987). On some geometric asp ects of Bruhat orderings II. The parab olic analogue of Kazhdan-Lusztig polynomials, J. Algebra. 111, 483-506 D. Deriziotis, T. P. McDonough and C. A. P allikaros (2010). On ro ot subsystems and involutions in S_n. Glasg. Math. J.. 52, 357-369 R. Dipp er and G. James (1986). Representations of Hecke algebras of general linear groups. Proc. London Math. Soc.(3). 52, 20-52 R. Dipper and G. James (1987). Blocks and idempotents of Hecke algebras of general linear groups. Proc. London Math. Soc.(3). 54, 57-82 M. Geck (2006). Kazhdan- Lusztig cells and the Murphy basis. Proc. London Math. Soc.(3). 93, 635-665 M. Geck and G. Pfeiﬀer (2000). Characters of finite Coxeter groups and Iwahori-Hecke algebras. London Mathematical So ciety Monographs, New Series, The Clarendon Press, Oxford University Press, New York. 21 G. James and A. Kerber (1981). The representation theory of the symmetric group. Encyclop edia of Mathematics and its Applications, Addison-Wesley Publishing Co., Reading, MA. 16 D. A. Kazhdan and G. Lusztig (1979). Representations of Coxeter groups and Hecke algebras. Invent. Math.. 53, 165-184 D. E. Knuth (1973). The art of computer programming. sorting and searching, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.. 3 G. Lusztig (1981). On a theorem of Benson and Curtis. J. Algebra. 71, 490-498 G. Lusztig (1984). Characters of reductive groups over a finite field. Ann. of Math. Stud., rinceton University Press. 107 G. Lusztig (1985). Cells in aﬃne Weyl groups. Algebraic groups and related topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., North-Holland, Amsterdam. 6, 255-287 T. P. McDonough and C. A. Pallikaros (2005). On relations b etween the classical and the Kazhdan-Lusztig representations of symmetric groups and asso ciated Hecke algebras, J. Pure Appl. Algebra. 203, 133-144 T. P. McDonough and C. A. Pallikaros (2008). On subsequences and certain elements which determine various cells in S_n. J. Algebra. 319, 1249-1263 B. Sagan (2001). The symmetric group, representations, combinatorial algorithms and symmetric functions. Graduate Texts in Mathematics, pringer-Verlag, New York. 203 J.-Y. Shi (1986). The Kazhdan-Lusztig cells in certain aﬃne Weyl groups. Lecture Notes in Mathematics, Springer. 1179 N. Xi (1994). Representations of Aﬃne Hecke Algebras. Lecture Notes in Mathematics, Springer-Verlag, Berlin. 1587