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Register a new userFour families of Weyl group orbit functions of B_3 and C_3
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The properties of the four families of special functions of three real variables, called here C-, S-, S^s- and S^l-functions, are studied. The S^s- and S^l-functions are considered in all details required for their exploitation in Fourier expansions of digital data, sampled on finite fragment of lattices of any density and of the 3D symmetry imposed by the weight lattices of B_3 and C_3 simple Lie algebras/groups. The continuous interpolations, which are induced by the discrete expansions, are exemplified and compared for some model functions.
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Weyl-orbit functions have been defined for each simple Lie algebra, and permit Fourier-like analysis on the fundamental region of the corresponding affine Weyl group. They have also been discretized, using a refinement of the coweight lattice, so that digitized data on the fundamental region can be Fourier-analyzed. The discretized orbit function has arguments that are redundant if related by the affine Weyl group, while its labels, the Weyl-orbit representatives, invoke the dual affine Weyl group. Here we discretize the orbit functions in a novel way, by using the weight lattice. A cleaner theory results, with symmetry between the arguments and labels of the discretized orbit functions. Orthogonality of the new discretized orbit functions is proved, and leads to the construction of unitary, symmetric matrices with Weyl-orbit-valued elements. For one type of orbit function, the matrix coincides with the Kac-Peterson modular $S$ matrix, important for Wess-Zumino-Novikov-Witten conformal field theory.
Four types of discrete transforms of Weyl orbit functions on the finite point sets are developed. The point sets are formed by intersections of the dual-root lattices with the fundamental domains of the affine Weyl groups. The finite sets of weights, labelling the orbit functions, obey symmetries of the dual extended affine Weyl groups. Fundamental domains of the dual extended affine Weyl groups are detailed in full generality. Identical cardinality of the point and weight sets is proved and explicit counting formulas for these cardinalities are derived. Discrete orthogonality of complex-valued Weyl and real-valued Hartley orbit functions over the point sets is established and the corresponding discrete Fourier-Weyl and Hartley-Weyl transforms are formulated.
Four families of generalizations of trigonometric functions were recently introduced. In the paper the functions are transformed into four families of orthogonal polynomials depending on two variables. Recurrence relations for construction of the polynomials are presented. Orthogonality relations of the four families of polynomials are found together with the appropriate weight fuctions. Tables of the lowest degree polynomials are shown. Numerous trigonometric-like identities are found. Two of the four families of functions are identified as the functions encountered in the Weyl character formula for the finite dimensional irreducible representations of the compact Lie group Sp(4). The other two families of functions seem to play no role in Lie theory so far in spite of their analogous `good properties.
Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G2, are compared and described. Two of the four families (called here C- and S-functions) are well known, whereas the other two (S^L- and S^S-functions) are not found elsewhere in the literature. It is shown explicitly that all four families have similar properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space, and they are discretely orthogonal when their values, sampled at the lattice points F_M subset F, are added up with a weight function appropriate for each family. Products of ten types among the four families of functions, namely CC, CS, SS, SS^L, CS^S, SS^L, SS^S, S^SS^S, S^LS^S and S^LS^L, are completely decomposable into the finite sum of the functions. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.
The affine Weyl groups with their corresponding four types of orbit functions are considered. Two independent admissible shifts, which preserve the symmetries of the weight and the dual weight lattices, are classified. Finite subsets of the shifted weight and the shifted dual weight lattices, which serve as a sampling grid and a set of labels of the orbit functions, respectively, are introduced. The complete sets of discretely orthogonal orbit functions over the sampling grids are found and the corresponding discrete Fourier transforms are formulated. The eight standard one-dimensional discrete cosine and sine transforms form special cases of the presented transforms.
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