Parameters of Lie geometries

It contains lots of numbers, or, rather, polynomials in q, that give association scheme parameters v, k_i, pⁱ_jk for association schemes defined on the geometry G/P of cosets of a parabolic subgroup P of a finite group of Lie type G defined over the field GF(q), especially for maximal parabolics P.

Note: the numbering of diagram nodes is non-standard here.

Properties

Look at a random valency (k₃ in E_7,7): q¹¹ + 2q¹² + 4q¹³ + 7q¹⁴ + 10q¹⁵ + 13q¹⁶ + 16q¹⁷ + 17q¹⁸ + 17q¹⁹ + 16q²⁰ + 13q²¹ + 10q²² + 7q²³ + 4q²⁴ + 2q²⁵ + q²⁶. What properties can one observe? The sequence of coefficients is symmetric (palindromic) and unimodal. The polynomial factors as q¹¹(q⁷−1)(q⁵−1)(q⁴−1)(q³+1) / (q²−1)(q−1)².

Given these polynomials, one can ask all kinds of questions. Are they symmetric (palindromic)? Are they unimodal? ...

Such questions have been asked e.g. by R. Stanley, see Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Alg. Discr. Meth. 1 (1980) 168-184, and also Log-concave and unimodal sequences in algebra, combinatorics, and geometry, pp. 500-535 in Graph Theory and Its Applications: East and West, Ann. New York Acad. Sci. 576, 1989. E.g., Theorem 19 in this last paper says that the number v of vertices of G/P is symmetric and unimodal (as polynomial in q).

Exercise Show that all valencies k_i are symmetric and unimodal (as polynomial in q). Find their factorization.

Symmetry is easy. Unimodality follows from Stanley. The factorization is found from the degrees of a suitable Coxeter subgroup.

A very conspicuous property is that the valency for the oppositeness relation is a power of q. More generally one can prove that all eigenvalues of the oppositeness graph have squares that are powers of q.

(A. E. Brouwer, The eigenvalues of oppositeness graphs in buildings of spherical type, pp. 1-10 in: Combinatorics and Graphs, R. A. Brualdi, S. Hedayat, H. Kharaghani, G. B. Khosrovshahi, S. Shahriari (eds.), AMS Contemporary Mathematics Series 531, 2010.)

What about the parameters pⁱ_jk?

Look at two random values in E_7,7. We have p⁸₁₇ = q² + 2q³ + 4q⁴ + 4q⁵ + 4q⁶ + 2q⁷ + q⁸, and p⁸₁₈ = −1 − q² − q³ − q⁴ + 3q⁷ + 3q⁸ + 4q⁹ + 3q¹⁰ + 2q¹¹ + q¹². We see that pⁱ_1j behaves well for j ≠ i, but not for j = i. Since the k_i factor nicely, and k_i pⁱ_1j = k_j p^j_1i, one expects a nice factorization for pⁱ_1j for j ≠ i. Also, the three-term expression for pⁱ_1j becomes a two-term one, with positive terms only, if j ≠ i, while pⁱ_1i also has contributions (q−1)q^a.

Problem Conjecture and prove properties of pⁱ_jk.

Files

A_n

B_n

D_n

Exceptional types

Twisted types

Explanation Consider G and P with thin analogs W and X, so that G = BWB and P = BXB. Let R be the set of fundamental reflections, so that W = <R>. For r ∈ R we consider X = <R\{r}>, so that P will be a maximal parabolic. We give the number of cosets |W/X| and the number of double cosets |X\W/X|. Next, for each double coset XwX an index i (in 0..|X\W/X|−1), its shortest representative w (as a string between parentheses, the digits j indicate the fundamental reflections r_j), its size k_i = |XwX/X| (as a number in square brackets) and |PwP/P| (as a polynomial in q). The double cosets X = X1X and XrX are numbered 0 and 1. Next we consider the graph Γ on the cosets gP, where gP, hP are adjacent when h⁻¹g ∈ PrP and give the double coset diagram of Γ. For each double coset XwX (viewed as a node in the double coset diagram, and identified with the set of vertices PwP/P) we give the number pⁱ_1j of neighbours that any of its vertices (such as wP) has in any other double coset.
For example, the graphs will be strongly regular when the number of double cosets is 3, and then the k_i give 1, k, v−k−1, and the pⁱ_1j give λ, μ, etc.

Erratum

References

[BCN] A. E. Brouwer, A. M. Cohen & A. Neumaier, Distance-regular graphs, Springer, 1989.