Branching rules for Weyl group orbits of simple Lie algebras B(n), C(n) and D(n)

TL;DR

This paper provides explicit branching rules for Weyl group orbits of simple Lie algebras B(n), C(n), D(n), detailing how these orbits decompose when restricted to maximal reductive subalgebras, with comprehensive matrices and examples.

Contribution

It systematically derives and lists matrices for orbit reductions of Weyl groups of B(n), C(n), D(n) for all n<=8 and key algebra-subalgebra pairs, including numerous examples.

Findings

Explicit matrices for orbit transformations listed for n<=8

Branching rules for various algebra-subalgebra pairs derived

Numerous examples illustrating orbit decompositions provided

Abstract

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