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

Se p 19 93 COMBINATORIAL PROOFS OF CAPELLI ’ S AND TURNBULL ’ S IDENTITIES FROM CLASSICAL INVARIANT THEORY

Combinatorial Proofs of q-Series Identities

We provide combinatorial proofs of six of the ten q-series identities listed in [3, Theorem 3]. Andrews, Jiménez-Urroz and Ono prove these identities using formal manipulation of identities arising in the theory of basic hypergeometric series. Our proofs are purely combinatorial, based on interpreting both sides of the identities as generating functions for certain partitions. One of these iden...

Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities I. Generalizations of the Capelli and Turnbull identities

We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy–Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull’s Capelli-type identities for symmetric and antisymmetric matrices.

Combinatorial and Automated Proofs of Certain Identities

This paper focuses on two binomial identities. The proofs illustrate the power and elegance in enumerative/algebraic combinatorial arguments, modern machineassisted techniques of Wilf-Zeilberger and the classical tools of generatingfunctionology.

. PR ] 1 3 A ug 1 99 9 Central Limit Theorem for Local Linear Statistics in Classical Compact Groups and Related Combinatorial Identities

We discuss CLT for the global and local linear statistics of random matrices from classical compact groups. The main part of our proofs are certain combinatorial identities much in the spirit of works by Kac and Spohn.