Transformation Groups

, Volume 3, Issue 4, pp 321–336

Kazhdan-Lusztig polynomials and canonical basis


  • I. B. Frenkel
    • Department of MathematicsYale University
  • M. G. Khovanov
    • School of MathematicsInstitute for Advanced Study
  • A. A. KirillovJr.
    • Department of MathematicsMIT

DOI: 10.1007/BF01234531

Cite this article as:
Frenkel, I.B., Khovanov, M.G. & Kirillov, A.A. Transformation Groups (1998) 3: 321. doi:10.1007/BF01234531


In this paper we show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the groupSn coincide with the coefficients of the canonical basis innth tensor power of the fundamental representation of the quantum groupUq\(\mathfrak{s}\mathfrak{l}\)k. We also use known results about canonical bases forUq\(\mathfrak{s}\mathfrak{l}\)2 to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmannians), due to Lascoux-Schützenberger and Zelevinsky.

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© Birkhäuser 1998