Branching rules for the Weyl group orbits of the Lie algebra A(n)

TL;DR

This paper studies how Weyl group orbits of the Lie algebra A(n) can be decomposed into orbits of its subalgebras, providing explicit matrices for transformations in various cases up to n=8 and for infinite series.

Contribution

It provides explicit transformation matrices for Weyl group orbit reductions of A(n) Lie algebras into subalgebra orbits, covering all cases up to n=8 and key infinite series.

Findings

Matrices for orbit transformations listed for n<=8

Decomposition rules for specific algebra-subalgebra pairs

Numerous examples illustrating the reduction process

Abstract

The orbits of Weyl groups W(A(n)) of simple A(n) type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of A(n). Matrices transforming points of the orbits of W(An) into points of subalgebra orbits are listed for all cases n<=8 and for the infinite series of algebra-subalgebra pairs A(n) - A(n-k-1) x A(k) x U(1), A(2n) - B(n), A(2n-1) - C(n), A(2n-1) - D(n). Numerous special cases and examples are shown.