On Discretization of Tori of Compact Simple Lie Groups II

TL;DR

This paper extends the discrete orthogonality properties of special functions derived from compact simple Lie groups to new families, providing explicit descriptions of their orthogonality on finite lattice fragments across various dimensions and symmetries.

Contribution

It introduces and explicitly describes the discrete orthogonality of new $S^s$- and $S^l$-function families derived from Lie group characters, expanding previous results.

Findings

Explicit orthogonality relations for $S^s$- and $S^l$-functions.

Orthogonality holds on finite lattice fragments $F^s_M$ and $F^l_M$.

Results apply to any dimension $n \\geq 2$ and any root length symmetry.

Abstract

The discrete orthogonality of special function families, called $C$- and $S$-functions, which are derived from the characters of compact simple Lie groups, is described in Hrivn\'ak and Patera (2009 J. Phys. A: Math. Theor. 42 385208). Here, the results of Hrivn\'ak and Patera are extended to two additional recently discovered families of special functions, called $S^s-$ and $S^l-$functions. The main result is an explicit description of their pairwise discrete orthogonality within each family, when the functions are sampled on finite fragments $F^s_M$ and $F^l_M$ of a lattice in any dimension $n\geq2$ and of any density controlled by $M$, and of the symmetry of the weight lattice of any compact simple Lie group with two different lengths of roots.