A preorder-free construction of the Kazhdan-Lusztig representations of Sn, with connections to the Clausen representations
نویسندگان
چکیده
We use the polynomial ring C[x1,1, . . . , xn,n] to modify the Kazhdan-Lusztig construction of irreducible Snmodules. This modified construction produces exactly the same matrices as the original construction in [Invent. Math 53 (1979)], but does not employ the Kazhdan-Lusztig preorders. We also show that our modules are related by unitriangular transition matrices to those constructed by Clausen in [J. Symbolic Comput. 11 (1991)]. This provides a C[x1,1, . . . , xn,n]-analog of results of Garsia-McLarnan in [Adv. Math. 69 (1988)]. Résumé. Nous utilisons l’anneau C[x1,1, . . . , xn,n] pour modifier la construction Kazhdan-Lusztig des modules-Sn irreductibles dans C[Sn]. Cette construction modifiée produit exactement les mêmes matrices que la construction originale dans [Invent. Math 53 (1979)], mais sans employer les préordres de Kazhdan-Lusztig. Nous montrons aussi que nos modules sont relies par des matrices unitriangulaires aux modules construits par Clausen dans [J. Symbolic Comput. 11 (1991)]. Ce résultat donne un C[x1,1, . . . , xn,n]-analogue des résultats de Garsia-McLarnan dans [Adv. Math. 69 (1988)].
منابع مشابه
RELATIONS BETWEEN THE CLAUSEN AND KAZHDAN-LUSZTIG REPRESENTATIONS OF Sn
We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x1,1, . . . , xn,n] to construct irreducible Sn-modules. This construction produces exactly the same matrices as the Kazhdan-Lusztig construction [Invent.Math 53 (1979)], but does not employ the Kazhdan-Lusztig preorders. It also produces exactly the same modules as those which Clausen constructed using a different basis ...
متن کاملA preorder-free construction of the Kazhdan-Lusztig representations of Hecke algebras Hn(q) of symmetric groups
We use a quantum analog of the polynomial ringZ[x1,1, . . . , xn,n] to modify the Kazhdan-Lusztig construction of irreducible Hn(q)-modules. This modified construction produces exactly the same matrices as the original construction in [Invent. Math. 53 (1979)], but does not employ the Kazhdan-Lusztig preorders. Our main result is dependent on new vanishing results for immanants in the quantum p...
متن کاملKazhdan–lusztig Cells and the Murphy Basis
Let H be the Iwahori–Hecke algebra associated with Sn, the symmetric group on n symbols. This algebra has two important bases: the Kazhdan–Lusztig basis and the Murphy basis. While the former admits a deep geometric interpretation, the latter leads to a purely combinatorial construction of the representations of H, including the Dipper–James theory of Specht modules. In this paper, we establish...
متن کاملConfiguration spaces, FS-modules, and Kazhdan-Lusztig polynomials of braid matroids
The equivariant Kazhdan-Lusztig polynomial of a braid matroid may be interpreted as the intersection cohomology of a certain partial compactification of the configuration space of n distinct labeled points in C, regarded as a graded representation of the symmetric group Sn. We show that, in fixed cohomological degree, this sequence of representations of symmetric groups naturally admits the str...
متن کاملKazhdan-lusztig Cells
These are notes for a talk on Kazhdan-Lusztig Cells for Hecke Algebras. In this talk, we construct the Kazhdan-Lusztig basis for the Hecke algebra associated to an arbitrary Coxeter group, in full multiparameter generality. We then use this basis to construct a partition of the Coxeter group into the Kazhdan-Lusztig cells and describe the corresponding cell representations. Finally, we speciali...
متن کامل