Three generalizations of Weyl's denominator formula
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.