On the proof of the generalized cauchy identity

Abstract

In this thesis, we consider the combinatorial proof of the generalized Cauchy identity.We know that the left hand side of the generalized Cauchy Identity is a generating function of the set of 3-dimensional matrices and the right hand side is a generating function of the set of triples of generalized permutations. Since there is a one-to-one correspondence between the set of 3-dimensional matrices of nonnegative integer entries and the set of triples of generalized permutations, by using the Robinson-Schensted-Knuth algorithm, we can obtain two mappings from the set of triples of generalized permutations into the set of ordered triples $(P_1,P_2,P_3)$ of generalized Young tableaux. Though these two methods do not give a combinatorial proof of the generalized Cauchy identity, we formulate and prove several interesting properties of them.