Gaussian cubature arising from hybrid characters of simple Lie groups
Lie groups with two different root lengths allow two ‘mixed sign’ homomorphisms on their corresponding Weyl groups, which in turn give rise to two families of hybrid Weyl group orbit functions and characters. In this paper we extend the ideas leading to the Gaussian cubature formulas for families of polynomials arising from the characters of irreducible representations of any simple Lie group, to new cubature formulas based on the corresponding hybrid characters. These formulas are new forms of Gaussian cubature in the short root length case and new forms of Radau cubature in the long root case. The nodes for the cubature arise quite naturally from the (computationally efficient) elements of finite order of the Lie group.
Department of Mathematics and Statistics, University of Victoria, Victoria, BC., V8W 3R4 Canada
Centre de recherches mathématiques, Université de Montréal, C. P. 6128 – Centre ville, Montréal, H3C 3J7, Québec, Canada
Département de mathématiques et de statistiques, Université de Montréal, C. P. 6128 – Centre ville, Montréal, H3C 3J7, Québec, Canada
MIND Research Institute, 111 Academy Drive, Irvine, California 92617
E-mail: email@example.com, firstname.lastname@example.org, email@example.com
Keywords: Gaussian and Radau cubature, Jacobi polynomials, simple Lie groups, Weyl groups
MSC: 65D32, 33C52, 41A10, 22E46, 20F55, 17B22
It has long been known that the Chebyshev polynomials of the second kind are related to the representation theory of , and of course to efficient methods of numerical quadrature. In  it was shown that there is a considerable generalization of this theory based on the series of lattices of type (so that the original theory applied to the lattice of type and the representations of ). This generalization depended deeply on the Weyl groups of these lattices, but not particularly on the Lie groups associated with them. The resulting formulas, now for functions of variables, went under the name of cubature formulas.
In  the idea that there is a genuine Lie theoretical connection here was extended to create a theory that works for every simple compact Lie group . The theory is again based on the root lattices but now also incorporates the representation theory of these groups in a deeper way, and more importantly uses the elements of finite order in the corresponding Lie group to define the nodes at which the cubature formulae are evaluated. The representations and the elements of finite order are in a sort of duality, and this duality plays a vital role in what happens. With a slight Lie-theoretical twist in the definition of the degrees of multi-variable polynomials, the crucial polynomials, their nodes and the cubature formulas appear completely naturally out of the theory and in fact are optimal (called Gaussian) in their efficiency.
The Weyl group , which appears as a group of reflections in this theory, is of primary importance, notably its sign homomorphism which takes the sign for each of the reflections in the roots. It has long been known in the theory of orthogonal polynomials based on these reflection groups that in the cases where the simple Lie group has roots of two different lengths (namely for types ) there are, in addition, two hybrid sign functions which distinguish between reflections in long roots and reflections in short roots; that is, the sign function takes the value for each reflection in a long root (respectively short root) and takes the value on the reflections in the short (respectively long) roots.
In this paper we extend the ideas of Chebyshev polynomials, nodes, and cubature formulas to these hybrid situations. In principle the path should be straightforward, particularly since orthogonal polynomials and -series based on this type of hybrid symmetry have been well studied, e.g. . However, our theory depends on both the representations and the elements of finite order of the Lie group, and this somewhat intricate process requires making a number of correct decisions in how to define things to fit the new setting. In the end things work out as smoothly and as naturally as in , although for the long root case the cubature is slightly less efficient than in the Gaussian cubature of the standard and short root cases, being instead what is called Radau cubature.
The orientation of  was towards the approximation theory community since Gaussian formulas are rather rare and the Lie theoretical connections offer new and unexpected techniques for constructing them. In this paper, in addition to presenting the new results based on hybrid Weyl symmetry and simplifying the overall presentation of the ideas, the emphasis is more the other way around, aiming to introduce the Lie theoretical community to some new applications of simple Lie groups to approximation theory and cubature. It seems to us that there is more to be explored here, particularly the duality between elements of finite order and character theory.
We begin with a summary of the results of  and then introduce the ideas which lead to the new cubature formulas arising from the two new families of orbit functions.
Start with the polynomial ring . This is given the structure of a graded ring by assigning a degree (called the -degree, for reasons to be explained later) to each of the variables . The degree of a monomial is thus . Unlike the usual gradation, need not be equal to . The value of will ultimately be the rank of a compact simple Lie group (or its complex simple Lie algebra ) and the degree structure will be given by the coefficients of its highest co-root.
The main result can be stated as a quadrature formula, called in this subject a cubature formula because it is not restricted to one dimension. Fix any non-negative integer . Then for all of -degree not exceeding ,
The main point is that integration is replaced by finite summing, and the elements of over which the summation takes place are very easy to compute. Here and is a finite subset of , is a constant, is a special polynomial in which is positive valued on . All of these objects depend on the choice of . In the hybrid situation that we shall develop here, the variables and similarly are real valued.
The elements of actually arise from elements of finite order, but in this context they are called the nodes, and they have a number of special properties. Their number is exactly the dimension of the space of polynomials of -degree at most . Furthermore, an important part of the construction of this result is the introduction of special polynomials (related to characters and other -invariant functions on ) of -degree , which form an orthogonal basis of with respect to the inner product
which in view of (1) is if the -degrees of do not exceed . Now, the minimum number of nodes that could achieve such an orthogonal decomposition of these functions is the dimension of the space of polynomials of -degree at most , and that is exactly the number of elements in . This optimal situation is called Gaussian cubature .
The nodes are actually zeros of certain of these polynomials of degree . The region is the image of the interior of the fundamental region (or some modified version of it in the hybrid cases) under a certain polynomial map. In particular it is an open set with compact closure and boundary of measure .
If we move to the Hilbert space of square integrable functions on with respect to the inner product then every function has a Fourier expansion
equality here being in the usual sense. If the sum is truncated to then this is the best approximation to in the -norm using only polynomials of -degree at most .
In essence what we have been describing arises from a duality that exists between the characters of the representations of and the conjugacy classes of elements of finite order of . Let be a maximal torus of . Since all the maximal tori are conjugate and every conjugacy class of meets every one of them, every character of is defined entirely by its restriction to and every conjugacy class of elements of finite order has elements in . The relationship between and its Lie algebra restricts to the relationship between and its Lie algebra:
Here it is more convenient to let be the Lie algebra of because the Killing form is then positive definite on , where is the rank of . The kernel of this exponential mapping is the co-root lattice of , so . The -dual of in is the weight lattice .
The normalizer of in is always larger than itself, and the Weyl group is the group that represents this excess. acts on via conjugation and then as linear transformations on . The affine Weyl group is then the semi-direct product of , which acts on with acting as translations.
Let be a standard simplicial fundamental region for in , so that is generated by the reflections in the faces of and is generated by the reflections in the faces of that pass through the origin, see . The virtue of is that it perfectly parametrizes the conjugacy classes of : for each such class there is a unique element of for which lies in that class.
The characters on restrict faithfully to -invariant functions on , and the ring of all -invariant functions on is a polynomial ring in -variables generated by the characters of a set of so-called fundamental representations. This is the ring and the can be viewed either abstractly as variables or as actual characters corresponding to a system of fundamental weights. One particularly important -invariant function on is where is the basic skew-symmetric function that appears as the denominator of Weyl’s character formula. This is the of the cubature formula.
Via the exponential mapping the characters can be viewed as -invariant functions on . In this way we have the important mapping
The region is the image of the interior of under .
There are several points of possible confusion regarding the many functions that appear in the paper. First of all there are many functions, like , which have interpretations as functions both on and on . This is not particularly troublesome since and all these functions are clearly periodic on with respect to . Thus interpreting as a function on or is rather obvious.
The second is the transition from exponential sums to new coordinates in using characters (or hybrid characters) as new variables. This is the way in which the Lie theory translates over into a theory about polynomials where the cubature formulas are relevant. Rather than introduce new function names when we transition variables, we use different notation for the variables. Thus for functions on or the generic variable name is , whereas for the new polynomial variables the generic variable name is . When we deal with short and long root scenarios, as we mostly do in what follows, we use in the short root case, and similarly for the long case.
There remains to briefly introduce the elements of finite order of . Each conjugacy class of an element of finite order has a unique representative in . The set is the image under of the set of elements in that have adjoint order . Here is the Coxeter number of and by adjoint order we mean that the order of the element is in the adjoint representation of on itself (i.e. by conjugation). The full order of an element is a finite multiple of the adjoint order.
This finishes our brief tour of the constituents of the basic cubature formula.
The Weyl group is a subgroup of the orthogonal group of with respect to its canonical Euclidean structure arising from the Killing form, and in particular there is the sign homomorphism
with for all reflections. The fact that is generated by the reflections in the roots of the Lie algebra plays an essential role in elucidating the structure of simple Lie groups and their representations. Throughout, -skew invariant functions and polynomials play a key role, Weyl’s character formula being a typical example which expresses the characters (-invariant exponential sums) as ratios of -skew invariant exponential sums. In the case when the roots of the Lie algebra have two distinct lengths (called the short and long roots), there are two alternative hybrid sign homomorphisms: which is defined by taking the value on the reflections in short roots and the value on the reflections in long roots, and which does it the other way around. This gives rise to new hybrid invariants, skew invariant with respect to short reflections while being invariant with respect to long, or vice-versa. This leads to two new versions of each cubature formula, see (6.2) which say very much the same thing except that and a new function , all appear in short and long forms according to which hybrid symmetry is used. The effect is somewhat subtle: is only altered along its boundary, the set changes only by certain elements of finite order along the boundary of the fundamental region , and the polynomial ring is still the space of -invariant functions. However the interpretations of the variables in terms of characters and the function are significantly altered.
We establish the notation that we are using and recall some basic facts about simple Lie algebras. For more details, see for example .
3.1. Simple Lie algebras
Let be a simple complex Lie algebra of rank with corresponding simple and simply connected compact Lie group . Let be a maximal torus of and let be its Lie algebra, so that we have the exponential map (4). Let on the dual space of be defined from the Killing form by duality. The natural pairing of and is denoted by .
Let denote the set of roots of and let be a set of simple roots, hence also a basis of . We denote by the corresponding Cartan matrix with entries
Its determinant, denoted by , is the order of the centre of and is also the index of the root (co-root) lattice in side the weight (co-weight) lattice, see below.
We introduce the usual partial ordering on : if and only if is a sum of simple roots or . The highest root in with respect to this ordering is denoted . Its coordinates in the -basis are called the marks:
Let be the root lattice and weight lattice respectively. Then
where is the system of simple co-roots (which forms a basis in ) defined by
To these simple co-roots corresponds the system of co-roots , which is in fact the system of roots for the simple Lie algebra with Cartan matrix (although this algebra never makes any real appearance in what follows). We have the highest co-root in and giving the co-marks :
It is these co-marks that define the degree function on later.
The lattice has as a basis the set of fundamental weights which is dual to the co-root basis in the sense that
This is so called basis of that we will use.
We also have two lattices in denoted and . The co-root lattice is kernel of the exponential map (4) with -basis consisting of the . The co-weight lattice is the -dual of in and has as a basis the set of fundamental co-weights defined by
The relationships between the lattices and between the various root and weight bases and their co-equivalents described below are summarized in:
Here the times symbol is meant to indicate that and , as well as and , are in -duality with each other.
Finally we have the cone of dominant weights:
3.2. Affine Weyl group and its dual
The Weyl group acting on is generated by simple reflections in the hyperplanes
By duality, we have the action of on where the simple reflections on co-root side are given by
The affine Weyl group is the semi-direct product of and the translation group : . Equivalently, can be defined as the group generated by the simple reflections and the affine reflection given by
where is the highest root of .
The standard simplex in defined by
serves as a fundamental domain for the affine Weyl group. Its vertices are
where , , are the marks (6). Note that is the reflection in the hyperplane
3.3. Long and short roots
In dealing with the hybrid cases, we are only interested in the simple Lie algebras with two different lengths of roots:
The root system of such algebras consists of short roots and long roots , so
Similarly, we decompose the set of simple roots as where and . Our indexing of the simple roots is such that
Since and are stabilized by and span , they both form root systems in . Although we do not use the facts here, it is known that is the root system of a semisimple subalgebra of the simple Lie algebra belonging to and is the root system of a subjoined semisimple Lie algebra [7, 8], which is usually not a subalgebra of .
where denotes the semisimple Lie algebra, , ( factors). In (10) we use the isomorphisms and .
Define the set of positive short and positive long roots by respectively.
is a system of positive roots for where .
All systems of positive roots in any root system arise as for some in the span of . Now with being half the sum of the positive roots of , we have . Then . So is a positive root system. ∎
The highest long root of coincides with the highest root of . So, the coefficients of written in basis are the marks , , see Table 1. The highest short root of denoted is given by its coefficients in basis, , see Table 1.
The dual root system decomposes also as disjoint union of short co-roots and long co-roots . The dual of is the highest short co-root . Note: we label the highest short root with ‘l’ to express the duality with the highest long root. Similarly, the dual of is the highest long co-root . The values of and are written in Table 1.
for which for is called a multiplicity function . The trivial example is for all which we denote simply by . Relevant for us are
we see that in addition to the usual half-sum of the positive roots we have
To and correspond the important short and long Coxeter numbers and defined by
The explicit calculations using the values in Table 1 imply that
4. invariant and skew invariant functions on
4.1. Sign homomorphisms
In addition to the usual sign homomorphisms on the Weyl group there are two others. This is well known, but since it is short we prove it. An abstract presentation determining is
where according as nodes and in the Coxeter-Dynkin diagram are not joined, joined by a single bond, a double bond, or a triple bond. Any homomorphism is determined by the values on the generators , . The necessary and sufficient condition for to be a homomorphism is that for all . This is automatically satisfied if is even. When is odd, i.e. , we need . Looking at the Coxeter-Dynkin diagrams we see that this allows precisely one choice of sign for all the short reflections and one for all the long reflections, and no other. Note that it does not matter whether or not we have a reflection in simple root or in any root since for any two roots of the same length there exists such that which implies . Thus there are four homomorphisms :
We shall use all four homomorphisms to introduce various classes of orbit functions.
4.2. and functions
Here the parameter is a dominant weight, the variable , is the orbit of , and where . Then is the number of points in where denotes the order of the Weyl group and is the number of points in the stabilizer in of . For functions, the summation is in fact over the whole of since has a trivial stabilizer.
When there are two different root lengths there are two other orbit functions, arising from the homomorphisms and :
where are given by (14). Here again we are defining for such that and for such that . This makes sense because the stabilizer in of is generated by long reflections , so takes the constant value on the stabilizer. Similarly, in (18) and are well defined.
Evidently the -functions are invariant while the (respectively , )-functions are (respectively , )-skew invariant.
All of these functions can be viewed as functional forms of formal exponential sums from of all linear combinations of formal exponentials with . In fact they are in since all the coefficients are integers. We write (respectively , ) for the invariant (respectively , -skew invariant) exponential sums, and similarly for the corresponding integral forms. More about the relationship between the formal exponentials and their use as functions may be found in .
The functions of , as we have defined them are functions on . However, since they are periodic modulo , they may be considered as functions on . This is the way in which we shall normally think of them. For integration purposes, an integral over rewrites to an integral over a fundamental domain for the lattice , for instance .
For notational convenience we use
which for each weight combines the exponential mapping of to and the -mapping on . As we have just said, we may think of as a function on .
We note specially that the and functions are sums over orbits rather than sums over the entire Weyl group. Obviously they can be rewritten as Weyl group sums, but in general these are redundant and for what follows the orbit sums are what we need. They also may be interpreted as functions on since they are invariant by -translations.
We show the result for , the proof for is similar. Let denote the Weyl group generated by short reflections and the Weyl group generated by long reflections. Then can be written as a semi-direct product where is a subgroup of . We know that the stabilizer of is generated by long reflections, so and
Thus the result is simply the usual formula that holds for all root systems. ∎
We are especially interested in the hybrid-characters:
They are clearly -invariant and we shall see that their linear span is . In particular they are well defined functions on all of (and, of course, they can be considered as functions on ). The hybrid characters for the fundamental weights also generate as a ring, and the main point is that they will become the new variables and . In fact these hybrid characters are in and and what we just said applies at the level of these rings. These facts are well known, but because of their central importance here we sketch out the proofs in what follows.
Inclusions in one direction are obvious. We show the reverse inclusion in the short case. Let and write . Let . Then and so, .
Thus we can divide into pairs where , , and (if then , so ). Thus for some finite subset .
Since and is always a factor of , we obtain for some ; and this statement is true for every . Now using  Ch.6, we have that are all relatively prime, and hence from for each we obtain . The result now follows. ∎
4.3. Domains et
The -functions are -skew invariant and are also translationally invariant with respect to . As such they are determined entirely by their restriction to the fundamental region . Because of Props. 4.1 and 4.2, the -functions vanish on the root hyperplanes of that correspond to the short roots, namely on . Define . We shall be interested in the -functions and their corresponding hybrid characters on this new domain.
All this can be done for the -functions too, and we define and . Note that the hyperplane appears in this case, since it is always associated with reflection in a long root.
Using (7), the domains and can be described by
Although and are proper subsets of , it is more relevant that each of them is a proper superset of . The original domain arises as a continuous image of via the mapping (5). The corresponding domains in the hybrid cases arise from in a similar way from these two supersets:
These will appear when we switch from variables to variables .
4.4. Jacobi polynomials
All the characters , the hybrid characters , , and the -functions , lie in . Furthermore each set forms a -basis for it and in each case the characters or hybrid characters indexed by the fundamental weights , , generate as a polynomial ring. Of course these facts apply to as well. This is quite easy to see because it is obvious that the -functions , , are a -basis for and the others can be written as sums of the form
where the . This triangular form with unit diagonal coefficients can be inverted in , showing that each of the other sets is a basis too. Similarly each can be written in the form
with integer coefficients, and this provides the recursive step to write any element of as a polynomial in the . The same thing can be done with the fundamental characters or hybrid characters.
Although we have no need for the specific values of the coefficients in these expressions, there are ways to compute them. As a specific example there are the Jacobi polynomials , defined for any multiplicity function , see , and any by
where the coefficients are defined recursively by: