# A preorder-free construction of the Kazhdan-Lusztig representations of Hecke algebras $H_n(q)$ of symmetric groups

Abstract : We use a quantum analog of the polynomial ring $\mathbb{Z}[x_{1,1},\ldots, x_{n,n}]$ to modify the Kazhdan-Lusztig construction of irreducible $H_n(q)$-modules. This modified construction produces exactly the same matrices as the original construction in [$\textit{Invent. Math.}$ $\textbf{53}$ (1979)], but does not employ the Kazhdan-Lusztig preorders. Our main result is dependent on new vanishing results for immanants in the quantum polynomial ring.
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Charles Buehrle, Mark Skandera. A preorder-free construction of the Kazhdan-Lusztig representations of Hecke algebras $H_n(q)$ of symmetric groups . 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.581-592. ⟨hal-01186303⟩

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