An umbral calculus for polynomials characterizing U( n) tensor operators

Abstract

After the change of variables Δ i = γ i − δ i and x i, i + 1 = δ i − δ i + 1 we show that the invariant polynomials μ G ( n) q (, Δ i , ; , x i, i+1 ,) characterizing U( n) tensor operators 〈 p, q,…, q, 0,…, 0〉 become an integral linear combination of Schur functions S λ ( γ − δ) in the symbol γ − δ, where γ − δ denotes the difference of the two sets of variables { γ 1 ,…, γ n } and { δ 1 ,…, δ n }. We obtain a similar result for the yet more general bisymmetric polynomials m μ G ( n) q ( γ 1 ,…, γ n ; δ 1 ,…, δ m ). Making use of properties of skew Schur functions S λ ρ and S λ ( γ − δ) we put together an umbral calculus for m μ G ( n) q ( γ; δ). That is, working entirely with polynomials, we uniquely determine m μ G ( n) q ( γ; δ) from m μ G ( n) q − 1 ( γ; δ) and combinatorial rules involving Ferrers diagrams (i.e., partitions), provided that n ≥ ( μ + 1) q. (This restriction does not interfere with writing the general case of m μ G ( n) q ( γ; δ) as a linear combination of S λ ( γ − δ).) As an application we deduce “conjugation” symmetry for n μ G ( n) q ( γ; δ) from “transposition” symmetry by showing that these two symmetries are equivalent.