# Parameters of Lie geometries

In this directory one finds a copy of the 1983 note Computation of some parameters of Lie geometries by A. E. Brouwer & A. M. Cohen. (PDF)

It contains lots of numbers, or, rather, polynomials in q, that give association scheme parameters v, ki, pijk 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 E7,7): 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 ki 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 pijk?

Look at two random values in E7,7. 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 pi1j behaves well for j ≠ i, but not for j = i. Since the ki factor nicely, and ki pi1j = kj pj1i, one expects a nice factorization for pi1j for j ≠ i. Also, the three-term expression for pi1j becomes a two-term one, with positive terms only, if j ≠ i, while pi1i also has contributions (q−1)qa.

Problem Conjecture and prove properties of pijk.

## Files

We give text files with parameters (kr and pijr, where r is a fundamental reflection) for some small cases.

### An

 A2,1 A3,1 A3,2 A4,1 A4,2 A5,1 A5,2 A5,3 A6,1 A6,2 A6,3 A7,1 A7,2 A7,3 A7,4

### Bn

 B2,1 B3,1 B3,2 B3,3 B4,1 B4,2 B4,3 B4,4 B5,1 B5,2 B5,3 B5,4 B5,5 B6,1 B6,2 B6,3 B6,4 B6,5 B6,6 B7,1 B7,2 B7,3 B7,4 B7,5 B7,6 B7,7

### Dn

 D4,1 D4,2 D5,1 D5,2 D5,3 D5,4 D6,1 D6,2 D6,3 D6,4 D6,5 D7,1 D7,2 D7,3 D7,4 D7,5 D7,6

### Exceptional types

 E6,1 E6,2 E6,3 E6,6 E7,1 E7,6 E7,7 E8,1 E8,7 E8,8 F4,1 F4,2 G2,1

### Twisted types

 3D4,1 3D4,2 2E6,1 2E6,2 2E6,3 2E6,4

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 rj), its size ki = |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 pi1j 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 ki give 1, k, v−k−1, and the pi1j give λ, μ, etc.

## Erratum

Comparison of the programs generating the above tables with the formulas in [BCN], shows a difference for twisted groups. In fact Proposition 10.7.4 ([BCN], p. 337) should, in the formula for pijr, have a final factor (q|r|−1) instead of (q−1), where |r| is the length of r as word in the untwisted Weyl group, that is, the length (1, 2, 3, or 4) of the longest word in the Weyl group generated by the fundamental reflections that map to r under the diagram twist.

## References

[BC] A. E. Brouwer, A. M. Cohen, Computation of some parameters of Lie geometries, Math. Centre Report ZW198, Sept. 1983.

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