Three generalizations of Weyl's denominator formula (Q1918867)

Summary: We give combinatorial proofs of three identities, each of which generalizes Weyl's denominator formula for two of the three root systems \(B_n\), \(C_n\), \(D_n\). Two of the three identities are due to S. Okada; the third appears in the author's doctoral thesis, upon which this work is based. Each of the identities we prove has a ``sum side'' and a ``product side''; both sides are polynomials in several commuting indeterminates. We use weighted digraphs to represent the terms on each side; the set of such digraphs that corresponds to the sum side is a proper subset of the set corresponding to the product side.