July 1996

by Mark Goresky

School of Mathematics

Institute for Advanced Study

Princeton NJ 08540

The Kazhdan-Lusztig polynomial P(y,w) may be defined for any pair of elements y,w in a group W generated by reflections. If W is the Weyl group of a semisimple Lie group G, then P(y,w) is the Poincare polynomial of the local intersection cohomology groups (in even degrees) of the Schubert variety associated with y at any point in the Schubert variety associated with w. Kazhdan and Lusztig described a general algorithm for the computation of P(y,w), which was later simplified and interpreted geometrically by S. Gelfand and R. MacPherson.

In 1981 I wrote a Pascal program to compute Kazhdan-Lusztig polynials and collated the results for A3, A4, A5, B3, B4, D4 and H3 in a Northeastern University preprint. These tables are available in hardcopy and also in electronic version. Since then, more efficient algorithms have been implemented by various people, especially by Fokko Ducloux.

The current tables consist of Kazhdan-Lusztig polynomials for the Weyl groups of type A3, A4, A5, C3, C4, D4, H3, and for Affine Weyl groups A2 tilde, A3 tilde, A4 tilde , B2 tilde, B3 tilde, C3 tilde, G2 tilde .

Some higher rank cases may be added in the future. In the affine case, each table is approximately 3000 lines long. Despite the relative inefficiency of the program, advances in computer hardware have made it very easy to extend these computations further and I will be happy to do so on request.

The tables are in ASCII format. I suggest you print them using a fixed spacing font such as COURIER. This way, the coefficients of the polynomials will appear in columns according to their degree. It makes the tables easy to search and read.

(1) D. Kazhdan and G. Lusztig. Representations of Coxeter Groups and Hecke Algebras. Inv. Math. 53 (1979), 165-184

(2) D. Kazhdan and G. Lusztig. Schubert varieties and Poincare duality. In: Geometry of the Laplace Operator. Proc. Symp. Pure Math. 36. Amer. Math. Soc. (1980), 185-203.

(3) S. Gelfand and R. MacPherson. Verma modules and Schubert cells: a dictionary. Seminaire d'algebre Paul Dubriel et M. P. Malliavin. Lecture Notes in Mathematics no. 925 (1982), 1-50. Springer Verlag.

(4) A. Lascoux and M. P. Schutzenberger: Polynomes de Kazhdan-Lusztig pour les Grassmanniennes. in: Young tableaux and Schur functors in algebra and geometry. Asterisque no. 87-88 pp. 249-266. Soc. Math. France, 1981.

The elements w of the reflection group W are listed in order of increasing length and are designated by an index number which appears in the lefthand column of each page. The "code" for each element w is a minimal expression of that element as a word in the simple reflections 1,2,...,r. The simple reflections correspond to the vertices of the Dynkin diagrams, as listed below. In the affine case, the generators 1,2,...,r-1 generate the (finite dimensional) Weyl group and the last generator r is the affine reflection.

In some of the tables there is a line that begins "codim 1" (It is my intention to re-run the program, printing out this information on all the tables, but at the moment it is only available on some tables). For a given Schubert cell v, this line contains a list of all the Schubert cells w such that w < v in the strong Bruhat order, and so that l(v) = l(w)+1. In other words it is a list of those Schubert cells w such that the closure of the cell v contains the cell w, and so that the complex dimension of w is one less than the complex dimension of v. The cells w are listed according to their "index" (an integer between 1 and |W|, the order of the Weyl group)

If the Schubert variety v is (homologically) nonsingular then the even Betti numbers of the variety are listed on the next line: the j th entry is the number of elements v of length j such that v < w in the strong Bruhat order (i.e. the number of j dimensional Schubert cells v which are in the closure of w). If the Schubert variety is singular then the next paragraph contains a list of its singularities:

SI@ v CODE c L= l KL: p

indicates that the Schubert variety w has a singularity along the Schubert cell with "index number" v. The code (c) and length (l) of v are reprinted for convenience and are followed by the coefficients of the Kazhdan Lusztig polynomial p=P(w,v). If v' < v and if P(w,v)=P(w,v') then the K.L. polynomial P(w,v') is not listed. Thus, for any v' < w, the polynomial P(w,v') will appear in the table, listed as P(w,v'') where v'' is the lowest entry on the list of singularities, such that v' is contained in the closure of v''. If it happens that several singularities with the same K.L. polynomial are printed, then these singularities are not comparable in the strong Bruhat order.

The list of singularities is followed by the even Betti numbers of the Schubert variety w, and by the intersection homology Betti numbers of w, arranged in order of increasing geometric dimension.

The polynomial 1 + q + 2q**2 is represented as KL: 1 1 2 and indicates that the local intersection cohomology group of w at a point in v is Q in degree 0 and 2, and is Q+Q in degree 4.

The following diagrams should be used to identify the way in which the generators were labelled in these tables.

A3: 1------2------3 A4: 1------2------3------4 A5: 1------2------3------4------5 C3: 1======2------3 C4: 1======2------3------4 D4: 1------2------3 | | 4 (5) H3: 1------2------3 /---3---\ A2tilde: / \ 1-------------2 /---4---\ / \ A3tilde: / \ 1--------2--------3 /--------5-------\ / \ A4tilde: / \ 1--------2--------3--------4 B2tilde: 3========1========2 B3tilde: 1 \ \ 2========3 / / 4 C3tilde: 4========1--------2========3 (6) G2tilde: 1--------2--------3