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Capelli's identity plays a prominent role in Weyl's approach to Classical Invariant Theory. Capelli's identity was recently considered by Howe and Howe and Umeda. Howe gave an insightful representation-theoretic proof of Capelli's identity, and a similar approach was used to prove Turnbull's symmetric analog, as well as a new anti-symmetric analog, that was discovered independently by Kostant and Sahi. The Capelli, Turnbulll, and Howe-Umeda-Kostant-Sahi identities immediately imply, and were inspired by, identities of Cayley, Garding, and Shimura, respectively. In this paper, we give short combinatorial proofs of Capelli's and Turnbull's identities, and raise the hope that someone else will use our approach to prove the new Howe-Umeda-Kostant-Sahi identity.

Combinatorial Proofs of Capelli's and Turnbull's Identities from Classical Invariant Theory