Mask formulas for cograssmannian Kazhdan-Lusztig polynomials

We give two contructions of sets of masks on cograssmannian permutations that can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the Iwahori-Hecke algebra. The constructions are respectively based on a formula of Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky. The first construction relies on a basis of the Hecke algebra constructed from principal lower order ideals in Bruhat order and a translation of this basis into sets of masks. The second construction relies on an interpretation of masks as cells of the Bott-Samelson resolution. These constructions give distinct answers to a question of Deodhar.Comment: 43 page

Relations between Young's natural and the Kazhdan-Lusztig representations of Sn

AbstractOur main result here is that, under a suitable order of standard tableaux, the classical representation of Sn introduced by Young (in “The Collected Papers of Alfred Young, 1873–1940,” Univ. of Toronto Press, Toronto) (QSA IV), and usually referred as the Natural representattion, the the more recently discovered (Invent. Math.53 (1979), 165–184) Kazhdan-Lusztig (K-L) representation are related by an upper triangular integral matrix with unit diagonal elements. We have been led to this discovery by a numerical exploration. We noted it in each of the irreducible representations of Sn up to n = 6. The calculations in these cases were carried out by constructing the corresponding Kazhdan-Lusztig graphs from tables (M. Goresky, Tables of Kazhdan-Lusztig polynomials, unpublished) of K-L polynomials. To extend the calculations to n = 7 we have used graphs obtained by means of an algorithm given by Lascoux and Schützenberger (Polynomes de Kazhdan & Lusztig pour les Grassmanniennes, preprint). Remarkably, the same property holds also for these graphs. These findings appear to confirm the assertion made by these authors that their algorithm does indeed yield K-L graphs. For the case of hook shapes we have obtained an explicit construction of the transforming matrices, a result which was also suggested by our numerical data. For general shapes, the transforming matrices are less explicit and our proof is based on certain properties of the Kazhdan-Lusztig representations given in their article (Invent. Math.53 (1979), 165–184) and on a purely combinatorial construction of the natural representation