On Ellipsoids Attached to Root Systems

For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.


Introduction
For any complex semisimple Lie algebra, there are a number of mathematical objects that are traditionally attached to it, and which determine it to some extent.The most widely used mathematical objects are: the Dynkin diagram, the Cartan matrix, the system of positive roots, the system of simple roots, the Weyl group, the universal enveloping algebra, etc.These objects have proved their usefulness in dealing with complex semisimple Lie algebras, and most of them have been generalized in order to deal with the new classes of mathematical structures, such as Kac-Moody algebras, superalgebras, quantum groups and Coxeter systems.
In this paper, two alternative mathematical objects are defined for any complex semisimple Lie algebra G .These objects are ellipsoids in the real linear space n  , where n is the rank of G .Given a complex semisimple Lie algebra G and a Cartan subalgebra H , the pair ( G , H ) determines the system of roots ∆ , a subsystem + ∆ of all positive roots, and the subsystem Π of all simple roots, see [1] [2].
In the space n  , define an inner product < , > in such a way that 2 , , , where ij a is an element of the Cartan matrix A defined by the system Π of simple roots, and i e is an element of the standard basis of n  , which we identify with the simple root i α .
Denote by n +  the subset of n  consisting of all that will be used in this paper.We assume the linear space n  to be partially ordered as follows: x y ≤ if and only if for any 1 i n ≤ ≤ we have i i x y ≤ .

Primary and Secondary Ellipsoids
The principal object of study in this paper is the subset of n  defined by the equation , 2 0.
This equation determines an ellipsoid in the space n  with the center at the point δ , and with the extreme points 0 and 2δ .We shall call this ellipsoid the primary ellipsoid and denote it by ( ) PE G .For the case of G belonging to the class 1 A , Equation (3) acquires the form 2 0, and so the primary ellipsoid in this case is the two-point subset {0,1} of 1  .For G belonging to the class 1 which is the equation of a circle passing through the points (0,0), (0,1), (1,0), and (1,1) of 2  .In cases of 2 A , 2 B , and 2 G , Equation (3) turns to be with k = 1 for 2 A , k = 2 for 2 B , and k = 3 for 2 G , which in all the three cases is equation of an ellipse passing through the points (0,0), (0,1), (1,0), (1,2), and 2δ .
In general case, the easiest way to write down Equation (3) in coordinate form is through the Dynkin diagram for the semisimple Lie algebra G , [2] [3].The Dynkin diagram has n vertices i a .Each vertice has a weight denoted by i k , which is an integer equal to 1, 2, or 3. Some of the vertices are connected by links, the number of edges in a link can also be equal to 1, 2, or 3.For any link connecting verices i a and j a , let ( , ) where the first sum is taken over all the vertices, and the second sum is taken over all the links in the Dynkin diagram for the complex semisimple Lie algebra G .Proof.By direct substitution of the vector F ; in all the other cases the coefficients ij l are equal to the number of edges in the link connecting the corresponding vertices.Owing to the fact that the Cartan matrix is positive definite, Equation ( 7) is equation of an ellipsoid in the space n  .This ellipsoid contains the origin 0 (0,..., 0) = and all points (0,.belongs to the primary ellipsoid ( ) E G defined by Equations ( 3) or (7).Proof.It is sufficient to show that the element m α α satisfies Equation (3).
Such x i h is a unique real number.Consider the vector 1 2 ( , ,..., )
Proof.By direct substitution of ( ) i T x with x i h evaluated by formula 10 into Equation ( 7) of the primary ellipsoid ( ) PE G .□ Observe that if x is an integral vector, that is a vector with all integer components, then x h is an integral vector as well.The set ( ) SE G is the set of all such vectors x n h ∈  as x runs through the primary ellipsoid ( )  is an ellipsoid, which is described by the equation or, equivalently, Proof.By direct calculation.□ In coordinate form, for the case 1 A , the equation of secondary ellipsoid has the form: So, in this case, the secondary ellipsoid is the two-point subset {1, 1} − of the real line.For G belonging to the class 1 1 A A ⊕ , the equation of secondary ellipsoid becomes which is the equation of the circle centered at the origin and passing through the points (1,1), and (−1, −1).In cases of 2 A , 2 B , and 2 G , the equation of secondary ellipsoid in coordinate form turns to be with k = 1 for 2 A , k = 2 for 2 B , and k = 3 for 2 G , respectively, which in all the three cases is the equation of an ellipse passing through the points (1,1), and (−1, −1), and with the center at the origin.
The primary ellipsoid is determined by the secondary ellipsoid in accordance with the formula 1 (1 ).

The equations of secondary ellipsoids in coordinate form for the simple Lie algebras of the four infinite series n
A , n B , n C , and n D are as follows:

h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h
4( 1) 2( 1) 8( 1) 4( 1) 6( 1) 0.

h h h h hh h h h h h h h h h h
Proof.By direct calculation.□ Corollary 2.1.A case-by-case examination of the equations of secondary ellipsoids in coordinate form (17)-(25) has shown that these equations can be written in a unified form as follows: where i k is the weight of the i-th vertice in the Dynkin diagram, and , ii ij b b are elements of the matrix 1 A − (the inverse matrix for the Cartan matrix A).
It is clear that this equation is valid for any complex semisimple Lie algebra, and not just for the simple Lie algebras.By multiplying Equation (26) with detA , we get an equation with all coefficients being nonnegative integers.

Diophantine Equations Derived from the Equations of Ellipsoids
Consider Equation ( 7) of primary ellipsoid and Equation (26) of secondary ellipsoid as Diophantine equations.This means that we are now concerned with only those solutions to these equations, which have all their components integers.We shall now explore the sets of all solutions of these Diophantine equations.These sets are nonempty and finite.We shall denote them by ( ) PD G and ( ) SD G respectively.The set ( ) PD G contains the elements 0 = (0, ••• ,0) and 2δ , and all the standard basis elements (0, ,1,., 0) i e = , and also 2 i e δ − , and even more, as follows from Proposition 2.2, for any positive root α , the element m α α , with m α being the grade of the positive root α , also belongs to the set ( ) PD G .Formula 10 implies that for any ( ) x PD ∈ G the vector 1 x h Ax = −  belongs to ( ) SD G , and this assignment is injective.Although the extreme points of the primary ellipsoid ( ) PE G (0 and 2δ ) both belong to n +  , the primary ellipsoid is not competely in n +  ; there exists an open neighbourhood U of the origin (in the primary ellipsoid) all of whose elements, except the origin itself, have at least one strictly negative component.This conclusion follows from the form of Equation (7).Formula 9 implies that for any ( ) x PD ∈ G and for any positive integer i with 1 i n ≤ ≤ , the element ( ) ( ,..., , , ,..., ) ( ) G also belongs to ( ) PD G .This fact allows us to find all solutions of these Diophantine equations and also to establish some of their properties.
The mappings i T are involutions of the primary ellipsoid.In the group of all permutations of the primary ellipsoid, consider the subgroup ( ) WPE G generated by the mappings i T .Theorem 3.1.The group ( ) WPE G is isomorphic to the Weyl group of the Lie algebra G .Proof.Follows from Coxeter relations.□ Corollary 3.1.The subset ( ) PD G of the primary ellipsoid is invariant under the action of the group ( ) WPE G and it splits into orbits.The set of the orbits is one-to-one with the subset of the set of all integral solutions with all nonnegative components of the equation of the secondary ellipsoid.For any such a sollution h, the vector 1 (1 ) , is the unique minimal vector of the respective orbit under the partial ordering ≤ .
We parametrize the set of the orbits by their minimal elements.For any such vector h a x = , denote by ( ) a PD G the respective orbit.For example, the number of orbits for the case of simple Lie algebra of class 8 E is equal to 157.For the simple Lie algebras of small rank

( , , , , )
A A A B G there is only one orbit.There is only one integral solution h in the secondary ellipsoid with all nonnegative components that has all its components positive, and it is equal to 1  .All other such solutions have at least one component equal to 0. For any such h consider all those values of index 1 2 ... k i i i < < < for which the corresponding component is equal to 0. In the Weyl group W, let h W be the subgroup generated by the elementary reflections 1 2 , ,..., e e e .

Corollary 3.2. The number of elements in the respective orbit is equal to the number ( ) / ( ) h card W card W .
There is only one orbit with the number of elements equal to the order of the Weyl group.This orbit contains the origin 0 and all the vectors m α α for any positive root α , as well as the element 2δ together with all the elements 2 m α δ α

−
, where m α is the grade of the positive root α .We shall call this orbit the main orbit and denote it by 0 ( ) PD G .The corresponding subset of the secondary ellipsoid will be denoted by 0 ( ).SD G The vectors from 0 ( ) PD G do not have negative components, and the vectors of 0 ( ) SD G do not have zero com- ponents.

Primary and Secondary Geometric Realizations of the Weyl Group
Denote by ( ) W G the Weyl group of the complex semisimple Lie algebra G .In this section we are concerned with geometric realizations of the Weyl group related to the primary and secondary ellipsoids.We first realize it in a matrix form.For any simple root i e , the matrix corresponding to the reflexion i s generated by i e is determined as follows: take the i-th line of the matrix I A − and replace by it the i-th line in the matrix I. Denote the matrix thus obtained by i M .This matrix is the matrix of the simple reflection i s .The matrix group generated by the matrices The image a reflection s α by the mapping is the vector m α α , where m α is the grade of the positive root α . Proof.
The image of a simple reflection i s by the mapping P is the basis vector i e .Now, consider the image of the product From this equality we also get the formula for the image of the inverse element: So, we shall define ( ) PW G to be the set 0 ( ) PD G with the transfered operation * , and call it the primary realization of the Weyl group.In this realization of the Weyl group the identity element is the origin 0 (0,..., 0) = .Formulas 29 and 30 imply that ( ) PW G is a group isomorphic to the Weyl group ( ) W G .This realization has some interesting features.Let (0,...,1,..., 0) , where m α is the grade of the positive root α defined by Formula 8.In this case we have that One more property is about multiplication on the left by the element 2δ .For any element ( ), In particular, (2 ) (2 ) 0.
If i e and j e are orthogonal simple roots, which means that 0 ij a = , then .
e e e e * = + And even more, if positive roots α and β are orthogonal, then If positive roots α and β are not orthogonal, then ( Proof.This is obvious for those cases ( , , ) G F E where det 1.

A =
In all the other cases, the proof follows from a case-by-case consideration.□ The mapping w w S Awδ → = , being a bijection, transfers the group structure from the Weyl group ( ) MW G to the set 0 ( ) SD G thus producing the secondary geometric realization of the Weyl group ( ) W G , which we shall denote by ( ) SW G .In this realization of the Weyl group, the identity element is 1  , and the element of maximal length is 1 −  .

Orderings of the Weyl Group
The realization ( ) PW G of the Weyl group on the primary ellipsoid provides a partial ordering of this group that is inherited from the natural partial ordering of the linear space n  .A vector x y ≤ .We denote this ordering by x y ≤ and call it the primary ordering of the group ( ) PW G .In this ordering of the group ( ) PW G , there is a unique minimal element 0 (0,..., 0) = , and a unique maximal element 2δ .Formula 33 implies that for any ( ) b PW ∈ G and for any i the elements b and i e b * are comparable under the primary ordering; if 0 ≤ * , and the inequality reverses otherwise.This statement can be generalized, by using Formula 39, to the case when we take any positive root α and its grade m α , and consider the element a m There is another very important for different applications ordering for any Weyl group, which is called Chevalley-Bruhat ordering, see [4]- [9], and which we denote by  .To define the Chevalley-Bruhat ordering, we first need to define the length of an element w of a Weyl group.The element w can be written as a product of elementary reflections i s .This can be done in several different ways.The minimal number of factors in such a representation of w is called the length of w (notation ( ) l w ).An expression of w as a product of elementary reflections with the number of factors equal to the length of w is called a reduced expression.The ellipsoid geometric realizations of the Weyl group provide a way to find the length of any element w of any Weyl group and also the family of all its reduced expressions. implies that a b ≤ .In this ordering, 0 is the unique minimal element too, and 2δ is the unique maximal.As a matter of fact, the primary is in some cases strictly stronger than the Chevalley-Bruhat ordering on the Weyl group, as one can see, for example, from the case of 3 A .In this case, the Weyl group is isomorphic to the symmetric group 4 S , and the graph of the Chevalley-Bruhat ordering for this group is available in Fig. 2.4 of [4].When compared to the primary ordering, it can be seen that there are two cases of discrepancy bitween the two orderings for this Weyl group.In the case of the Chevalley-Bruhat ordering, the elements (1 4 3 2) and (4 1 2 3) of 4 S are not comparable, as can be seen from Fig. 2.4 of [4], but their respective counterparts in the primary ellipsoid geometric realization are the vectors (0,2,2) and (1,2,3), which are comparable in the primary ordering.The same holds true for the elements (3 2 1 4) and (2 3 4 1) of 4 S , which have the vectors (2,2,0) and (3,2,1) as their respective counterparts.In all the other cases, the two orderings agree for 3 A .Observe that in these two cases of a b ≤ but not a b  we have that (1, 0,1) b a − = , which is not a multiple of a positive root.
To Chevalley-Bruhat order a Weyl group W by using the primary realization take the following steps: 1) Realize W primarily by assigning ( ) P w w δ δ = − to any w; 2) Order the primary realization primarily by inserting a link between any two directly adjacent elements a b ≤ ; 3) Delete all those links with a b ≤ for which b a − is not a multiple of a positive root.The remaining links provide the Chevalley-Bruhat ordering of the Weyl group W.
The secondary realization of a Weyl group provides an efficient way to obtain all reduced expressions for any ellement w of the Weyl group.A reduced expression of w is a shortest possible expression of it as a product of simple reflections i s .Finding all reduced expressions of any element of a Weyl group boils down to finding the first element in any such expression, because if s is known to be the first element of a reduced expresion for w, to find the second element of this reduced expression is equivalent to finding the first element of the product sw, and so on.Proof.This follows directly from the definition of the secondary realization of the Weyl group.□ This theorem provides an alternative way to build the Chevalley-Bruhat ordering of a Weyl group, because, as is well known for any Coxeter group (see for example [8]), knowing reduced expressions leads to Chevalley-Bruhat ordering through subexpressions.

Corollary 4 . 2 .
The image of the Weyl group ( ) MW G by the mapping P is the main orbit 0 ( ) PD G .Motivated by formulas 29 and 30, define the group operation denoted by * on the set 0 ( ) PD G as follows.By Proposition 4.2, for any 0

G
being equal to a multiple of a positive root α , a m product a b α * and the element b are also comparable.

Theorem 5 . 1 .
Given an element w of the Weyl group( ) MW G , consider its image ( ) S w Awδ = in ( ) SW G .The vector ( )S w being an n-tuple of positive and negative integers 1 2 ( , ,..., ), only simple reflections that can be the first elements of a reduced expression of w.
Let δ be the half-sum of all positive roots in this realization of + +∆ can be considered as a subset of n +  .
We shall call the number m α the grade of the root α .
The proof follows from the fact that the Weyl group acts simply transitively on the set of all Weyl chambers.□ .