A curious identity and the volume of the root spherical simplex.
Abstract.
We show a curious identity on root systems which gives the evaluation of the volume of the spherical simpleces cut by the cone generated by simple roots. In the appendix John Stembridge gives a conceptual proof of our identity
1. Introduction
In this note we shall consider a finite root system spanning an euclidean space of dimension (for all the facts about root systems which we are going to use in this note we refer to [1]). is called the rank of . We shall choose once and for all a set of positive roots and in the set of simple roots . We shall also denote by the Weyl group of i.e. the finite group generated by the reflections with respect to the hyperplanes orthogonal to the roots in . Given such a root , we shall denote by the reflection with respect to the hyperplane orthogonal to . Set for each and call the set of simple reflections. One know that generates and that the pair is a Coxeter group.
We know that the ring of regular functions on , invariant under the action of , is a polynomial ring generated by homogenous elements of degrees . The ’s are called the degrees. We shall also consider the sequence of exponents, . Recall that .
In we have the affine arrangement of the hyperplanes orthogonal to the roots and their translates under the weight lattice , a locally finite configuration invariant under the affine Weyl group . is the semidirect product of and of the lattice spanned by the roots, thought of as translation operators.
is itself a Coxeter group. In the case in which is irreducible, its Coxeter generators are given by the reflections , where the for the ’s are the simple generators of and
being the longest root. One knows that, for each , the subgroup of generated by the reflections is finite, and it is the Weyl group of a root system which will be discussed presently. Hence we can consider the degrees . Our main result is the identity (Theorem 1.2)
The proof is a case by case computation using the classification of irreducible root systems. It is quite desirable to give a more conceptual deduction of our identity.
In the last section we show, following a suggestion of Vinberg, that our identity implies the following geometric identity. Take the unit sphere in and consider the spherical simplex , being the cone of positive linear combinations of the simple roots. Then
We have discovered this identity while trying to understand the following fact.
Consider the complex space , and take the algebraic torus . For any root the linear form defined by
takes integer values on , hence we get the character of .
Denote its kernel by . In our work on toric arrangements (see [2] ,[5], [6], [7], [8]) we have shown that the Euler characteristic of the open set equals . The only proof we know of this fact is via a combinatorial topological construction of Salvetti [4] [3]. The above identity has been the result of an attempt to give a direct computation of this Euler characteristic.
1.1. The main identity
We are interested in the numbers
The following table gives in the case of irreducible root systems
Notice that if is reducible, i.e. with , then clearly
We normalize the scalar product so that the short roots have length and we denote by the Dynkin diagram of .
From now on we assume that is irreducible and we denote by the extended Dynkin diagram. Set , with the highest root. We have a bijection between the set and the nodes of . For every the diagram obtained from removing the node corresponding to the root (and all the edges having that node as one of the vertices) is of finite type. So we can consider the corresponding root system consisting of all roots in which are integral linear combinations of the roots and the corresponding number .
During the proof of our result we shall need the following well known
Lemma 1.1.
The following identities hold,
(1) 
Furthermore, when we have:
(2) 
(3) 
Proof.
The first identity follows immediately from the following power series expansion
(4) 
To see this notice that setting
we have
from which we deduce that
Writing we deduce that
Equating coefficients, we get and for
On the other hand if we set , we get and
so and everything follows.
To see the second identity, notice that using (4) and integrating, we get
(5) 
Again using (4) we deduce
(6) 
This together with (4) implies that
is the coefficient of in the power series expantion of
since the claim follows. To see the last identity, let us remark that its left handside is the coefficient of in the power series expansion of the function
From this everything follows.
∎
Theorem 1.2.
Proof.
The proof is by a case by case computation.
Let us deal first with the exceptional cases. In order to make the computation transparent it is more convenient to multiply our sum by , the order of the Weyl group
Case . In this case . By looking at the extended Dynkin diagram
we get that
Case . The order of the Weyl group is . By looking at the extended Dynkin diagram
we get that
Case . In this case . By looking at the extended Dynkin diagram
we get that
Case . In this case . By looking at the extended Dynkin diagram
we get that
Case . In this case . By looking at the extended Dynkin diagram
we get that
Case . In this case each is of type . It follows that
Case . The extended Dynkin diagram is
we get that, denoting by the trivial root system and setting ,
by Lemma 1.1, part (1).
Case . The extended Dynkin diagram is
we get that, denoting by the trivial root system, setting , and ,
by Lemma 1.1, part (2).
Case . The extended Dynkin diagram is
we get that, setting and ,
which equals 1 by Lemma 1.1, part (3). ∎
1.2. The volume of
Recall that we have introduced the spherical simplex as the intersection of the unit sphere in with the cone of non negative linear combinations of the simple roots for the root system . Our purpose is to show
Theorem 1.3.
Proof.
For simplicity we normalize in such a way that . We then set . If is reducible, i.e. with , we have
Since we also have
an easy induction implies that we are reduced to show our claim under the assumption that is irreducible.
So assume irreducible and set , with the highest root. Write with a negative integer for all .
As in the previous section for every set equal to the root system consisting of all roots in which are integral linear combinations of the roots so that in particular . Recall that the Dynkin diagram of is the subdiagram of obtained by removing the node corresponding to . The roots are simple roots for .
We claim that is the union of the cones whose interior are disjoint. To see this take , write . If all are larger or equal than zero then , otherwise for at least one index . Take an index for which is maximum. Notice that necessarily . We can clearly write
and all coefficients are non negative.
Now observe that if, for any , we write as a linear combination of then all coefficients are negative. We then leave to the reader the easy verification that this implies that the interiors of the cones are mutually disjoint.
References

[1]
Bourbaki N., Groupes at algèbres de Lie, Chapitres 4,5,6 Actualités Scientifiques et Industrielles, No. 1337 Hermann, Paris 1968 288 pp.

[2]
De Concini, C.; Procesi, C. On the geometry of toric arrangements. Transform. Groups 10 (2005), no. 34, 387–422.

[3]
De Concini, C.; Salvetti, M. Cohomology of Artin groups: Addendum: ”The homotopy type of Artin groups” Math. Res. Lett. 3 (1996), no. 2, 293–297.

[4]
Salvetti, Mario The homotopy type of Artin groups. Math. Res. Lett. 1 (1994), no. 5, 565–577.

[5]
A. Henderson, The symmetric group representation on cohomology of the regular elements of a maximal torus of the special linear group, preprint http://www.maths.usyd.edu.au/u/anthonyh/

[6]
Lehrer, G. I. The cohomology of the regular semisimple variety. J.
Algebra 199 (1998), no. 2, 666–689.

[7]
Lehrer, G. I. A toral configuration space and regular semisimple conjugacy
classes. Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1,
105–113.

[8]
E. Looijenga, Cohomology of and . Mapping class groups and moduli
spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991),
Contemp. Math., 150, Amer. Math. Soc.,
1993, 205–228.
appendix
john r. stembridge ^{1}^{1}1 Work supported by NSF grant DMS–0245385.
In this appendix, we provide an explanation for the “curious identity” (Theorem 1.2) without any casebycase considerations. The proof is based on two elegant formulas, one due to L. Solomon, the other due to R. Steinberg. Both of these results deserve to be better known.
If is a finite group generated by reflections in a real Euclidean space , consider the class function on defined by
where the determinants are evaluated as endomorphisms of , and are indeterminates. This may be viewed as a bigraded character for , the tensor product of the symmetric and exterior algebras of .
In his 1963 paper on invariants of finite reflection groups [1], Solomon explicitly determined the structure of the invariants of . At the level of characters, his structure theorem implies
where are the degrees (), denotes the trivial character of , and is the usual pairing of realvalued class functions and .
Henceforth, assume that is a Weyl group with an irreducible root system of rank and simple reflections . Note that by setting and letting in (1), we obtain the quantity .
We let denote the reflection corresponding to the highest root and set . One may interpret as the image of the simple reflections of the associated affine Weyl group .
Following Steinberg (see Section 3 of [2]), the action of on descends to a action on the torus (where denotes the root lattice), and the decomposition of into simplicial alcoves by the arrangement of affine hyperplanes associated to induces a simplicial decomposition of with a compatible action. Moreover, the stabilizers of the faces of are (up to conjugacy) generated by the various proper subsets of .
Given , Steinberg computes the Euler characteristic of the fixed subcomplex of in two different ways (see Theorem 3.12 of [2]), thereby obtaining the identity
where denotes the reflection subgroup generated by , and denotes the permutation character of the action of on . It is important to note that ranges over proper subsets of .
Steinberg actually proves a more general identity that involves twisting by an involution; the above instance corresponds to the trivial involution. One may also recognize (2) as a companion to the more familiar identity
Now consider the evaluation of
First, notice that as , so we obtain
by setting in (1).
Second, notice that , so (2) implies
by Frobenius reciprocity. We can evaluate each of these terms by applying Solomon’s formula to the reflection group . But we need to be careful, because the action of on will have linear invariants if the rank of is less than . In such cases, this means that some of the degrees of will equal 1, which introduces factors of in (1). Since we have set , these factors vanish.
Thus (4) should be restricted to subsets of , and we obtain
where are the degrees of for . Comparing this with (3) in the limit , we obtain the “curious identity”
References

[1]
L. Solomon, Invariants of finite reflection groups,
Nagoya Math. J. 22 (1963) 57–64.
 [2] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. (1968), no. 80.