Poincare series for the depths of roots in a root system
Dmitrii V. Pasechnik
The depth of a positive root a in a root system F related to a Coxeter group G
is the minimal length of g in G such that the root g a is negative.
(Equivalently, if the refelection r_a has length l then depth a = (l+1)/2)
We conjecture that the function P(t)=a_1 t + a_2 t^2 + a_3 t^3 +..., where
a_i is the number of roots of depth i,
is always rational.
Some (computational) evidence to support this.
First,
For affine Coxeter groups a_i is a periodic function (of i) with a_i determined
in a simple way from a_i's for the associated finite group.
For instance, for affine A_n all a_i=n.
(this is not (yet) a theorem, but we suspect it's not so difficult to show.)
Second,
I give a (conjectural) recurrence relation for each case, and in the 1st case I
also write down the generating function.
(The latter is just a straightforward computation given the
recurrence. One could also compute a(n) as a function of n, although
it wouldn't look too nice.)
Note that for the sequence to satisfy a recurrence relation is
equivalent to have a rational generating function, cf. e.g.
R.Stanley "Enumerative Combinatorics I", Wadsworth 1986, Chapter 4.
1) o--o n: 1 2 3 4 5 6 7 8 9 10 11
|\/| 2*( 2 3 6 12 27 60 138 315 726 1668 3843....)
|/\|
o--o a(n)=2a(n-1)+2a(n-2)-3a(n-3).
F(x)=\sum a(n) x^n=2x((2-x-10x^2)/(1-2x-2x^2+3x^3)).
2) o--o 1 2 3 4 5 6 7 8 9 10 11
| /| 4 5 8 13 24 44 83 158 303 582 1120
|/ |
o--o a(n)=2a(n-1)-a(n-5).
3) o--o 1 2 3 4 5 6 7 8 9 10 11
| / 4 4 5 6 8 11 15 21 30 43 62
|/
o--o a(n)=2a(n-1)-a(n-2)+a(n-3)-a(n-4) (for n>5).
4)
o
|\
| o---o---o
|/
o
The depths are as follows:
[ 5, 5, 6, 8, 11, 16, 25, 38, 59, 93, 148, 235, 376, 602, 966, 1550, 2491,
4003, 6436, 10348, 16643, 26766, 43052, 69247 ]
The recurrence is as follows:
a(n+1)=\sum_{i=n-11}^n v(i)*a(i), for v=[0,0,-1,-1,-2,-1,0,2,2,1,0].
I would appreciate receiving any comments on this.
(please copy your reply to my email di...@win.tue.nl)
Dmitrii V. Pasechnik
Department of Mathematics
Eindhoven University of Technology
PO Box 513, 5600 MB Eindhoven
The Netherlands
e-mail: di...@win.tue.nl
http://www.can.nl/~pasec