Electronic J. Combinatorics, 1, 1994, R1
Dominique Foata and Doron Zeilberger
Combinatorial Proofs of Capelli's and Turnbull's identities from
Classical Invariant Theory
Abstract.
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 by Howe and Umeda 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.
foata@math.u-strasbg.fr, zeilberg@math.temple.edu
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