×
Im, Mee Seong

A short proof on the transition matrix from the Specht basis to the Kazhdan-Lusztig basis. (English) Zbl 1487.05273

Rocky Mt. J. Math. 51, No. 5, 1671-1680 (2021).
Summary: We provide a short proof on the change-of-basis coefficients from the Specht basis to the Kazhdan-Lusztig basis, using Kazhdan-Lusztig theory for the parabolic Hecke algebra.

MSC:

05E10 Combinatorial aspects of representation theory
20C08 Hecke algebras and their representations
20C30 Representations of finite symmetric groups

Keywords:

Kazhdan-Lusztig basis; Specht module; parabolic Hecke algebra
PDFBibTeX XMLCite
\textit{M. S. Im}, Rocky Mt. J. Math. 51, No. 5, 1671--1680 (2021; Zbl 1487.05273)
Full Text: DOI arXiv Link

References:

[1] H. Bao, J. Kujawa, Y. Li, and W. Wang, “Geometric Schur duality of classical type”, Transform. Groups 23:2 (2018), 329-389. · Zbl 1440.17009 · doi:10.1007/s00031-017-9447-4
[2] A. Björner and F. Brenti, Combinatorics of Coxeter groups, Grad. Texts in Math. 231, Springer, 2005. · Zbl 1110.05001
[3] R. Dipper and G. James, “Representations of Hecke algebras of general linear groups”, Proc. London Math. Soc. (3) 52:1 (1986), 20-52. · Zbl 0587.20007 · doi:10.1112/plms/s3-52.1.20
[4] A. M. Garsia and T. J. McLarnan, “Relations between Young’s natural and the Kazhdan-Lusztig representations of \[S_n\]”, Adv. in Math. 69:1 (1988), 32-92. · Zbl 0657.20014 · doi:10.1016/0001-8708(88)90060-6
[5] G. D. James, The representation theory of the symmetric groups, Lecture Notes in Math. 682, Springer, 1978. · Zbl 0393.20009
[6] G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications 16, Addison-Wesley, Reading, MA, 1981. · Zbl 0491.20010
[7] H. Naruse, “On an isomorphism between Specht module and left cell of \[\mathfrak S_n\]”, Tokyo J. Math. 12:2 (1989), 247-267. · Zbl 0761.20008 · doi:10.3836/tjm/1270133181
[8] H. M. Russell and J. S. Tymoczko, “The transition matrix between the Specht and web bases is unipotent with additional vanishing entries”, Int. Math. Res. Not. 2019:5 (2019), 1479-1502. · Zbl 1468.20025 · doi:10.1093/imrn/rnx164
[9] W. Soergel, “Kazhdan-Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln”, Represent. Theory 1 (1997), 37-68. · Zbl 0886.05124
[10] W. Specht, “Die irreduziblen Darstellungen der symmetrischen Gruppe”, Math. Z. 39:1 (1935), 696-711 · Zbl 0011.10301 · doi:10.1007/BF01201387
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.