Four types of special functions of
and their discretization
Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group , are compared and described. Two of the four families (called here - and -functions) are well known. New results of the paper are in description of two new families of -functions not found in the literature. They are denoted as - and -functions.
It is shown that all four families have analogous useful properties. In particular, they are orthogonal when integrated over a finite region of the Euclidean space. They are also orthogonal as discrete functions when their values are sampled at the lattice points and added up with appropriate weight function. The weight functions are determined for the new families. Products of ten types among the four families of functions, namely , , , , , , , , and , are completely decomposable into the finite sum of the functions belonging to just one of the families. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.
Discretization of characters of irreducible finite dimensional representations of any compact simple Lie group was introduced in general in [MP87]. At the time it was motivated by the need to solve large computational problems in Lie theory. A quarter of century later the motivation would rightfully be called processing of digital data from lattices of any dimension, density, and symmetry. Large computations made it a practical imperative to replace the basis of irreducible characters by the basis of constituents of the characters that are symmetric with respect to the Weyl group denoted here -functions. Unlike the characters, -functions are firmed as sums of limited number of exponential terms.
It is perhaps curious to note that such a discretization was made possible only after publication of three papers during the few years preceding [MP87]. They were the following: (i) a concise description all conjugacy classes of elements of finite order in any in [Kac], then (ii) after it became possible to decompose even the extremely large characters into the sum of the sum of constituent -functions [BMP, MP82], and finally (iii) after the computation of character values of elements of finite order in became practical [MP84]. The discretization of the skew-symmetric -functions in [MP06] follows an analogous path.
The - and -functions appear in the Weyl character formula [BS]. The functions have been known, for more than half a century, as different constituent of the character [Bourbaki]. They are well known for the uniformity of their properties across the compact simple Lie groups of any type, in particular their continuous orthogonality when integrated over a finite region of the real Euclidean space, and their discrete orthogonality when summed up over a lattice fragment in of any density [MP87, MP06]. In the specific case of , they were described in [PZ2] and [PZ3] respectively.
The aim of this paper is to confirm for the simple Lie group , the existence and properties of two additional families of special functions denoted and . Existence of such functions was first noticed111In [MotP] different notations are used. Namely , , and . in [MotP] for the group . All four families of functions are obtained by a summation of exponential terms over the Weyl group of . They are refereed to as orbit functions. In 1-dimension, when the underlying group is the rank one simple Lie group (equivalently ), the - and -functions become the common cosine and sine functions, while - and -functions have no analog there. We point out in the paper that the families of - and -functions do not exist for the group (equivalently ).
The history of the functions of the families and is short. It was noted in [MotP] that the 2-variable antisymmetric cosine and symmetric sine can be obtained by summation of appropriate exponential functions over the Weyl group of the simple Lie group . Since [MotP] is a special case of [KP4], where the -variable symmetric and antisymmetric sine and cosine were defined, the families of - and -functions could have been noted already in [KP4] in the context of the simple Lie groups of any rank.
The present paper is thus the first to describe the families and of orbit functions outside of the symmetric and antisymmetric generalizations of trigonometric functions, their symmetries are more complicated. Most recently it was shown that the four families of orbit functions exist for all compact simple Lie groups with two root lengths [MMP].
The new results of the paper, which are the properties of - and -functions of , are brought together and compared with the properties of the -orbit functions of all four families.
The second goal of the paper is to describe the properties of the new families of orbit functions when they are discretized, that is when their values are sampled on a lattice fragment . It turns out that the functions within families , , and are orthogonal when summed up over the same finite fragment of the lattice in of any density specified by . Although the same grid may be used for each family, the appropriate weight functions are different. That makes the functions of either family suitable for expansion of 2-dimensional digital data. This has been known for over two decades [MP87] and used in some very large computations, for example in [GP]. It is also known for -functions [MP06, PZ3]. For and types. It is exposed only here and in the forthcoming papers [MMP, MotP].
The definition of the orbit function is contained in Sec. LABEL:OF together with description of their behavior at the boundary of the domain and their continuous orthogonality in . It is shown that all the orbit functions are either real or purely imaginary. A simple renormalization makes the later functions also real.
In section LABEL:Discret, orbit functions are discretized. Discrete orthogonality of the orbit functions is proven.
In section LABEL:Decomp, the decomposition of the products of pairs of orbit functions into their sums is described. All 10 products, , , , , , , , , and , decompose into the sum of orbit functions from just one family. A product of a pair of orbit functions contains either 144 or 72 or 36 exponential terms which can be rewritten as half as many trigonometric function terms. The information is then used to find the recurrence relations for the orbit functions.
Arithmetic properties of the orbit functions remain mostly unexplored in the literature. In section 6, we present just the properties of -functions, directly deduced from the corresponding properties of the characters [MP84].
In there are precisely 14 (conjugacy classes of rational) elements at which all the -functions have integer values. In Table LABEL:efos we list the 14 elements specified by the values of their coordinates , and show the values of -functions at several lowest orbits of . Corresponding properties of the families , and are not known.
The two simple roots of and their numbering are shown on the Dynkin diagram
They span the real Euclidean space . Standard conventions for writing the diagram identify the lengths and relative angle between the simple roots as given by
Relations between the four bases we will use can be written explicitly:
They follow from the -duality requirement
The link between - and -bases is provided by the Cartan matrix of and its inverse,
The determinant of the Cartan matrix is 1, therefore the root lattice and the weight lattice coincide in the case of .
Reflections and in mirrors passing through the origin and orthogonal to simple roots generate the root system from the simple roots.
The set of positive roots , the set of positive long roots , the set of positive short roots and their half sums , , are the following: