Generalizations of Cauchy's determinant and Schur's Pfaffian

Abstract

We present several identities of Cauchy-type determinants and Schur-type Pfaffians involving generalized Vandermonde determinants, which generalize Cauchy's determinant det (1/(xi + yj)) and Schur's Pfaffian Pf ((xj − xi)/(xj + xi)). Some special cases of these identities are given by S. Okada and T. Sundquist. As an application, we give a relation for the Littlewood-Richardson coefficients involving a rectangular partition.

Pfaffians and representations of the symmetric group

“…Thus, the evaluation of the Pfaffian of order 2m has been reduced to the evaluation of a determinant of order m. Ishikawa [4], Okada [18], and Ishikawa-Okada-Tagawa-Zeng [5] have given many generalizations of Sundquist's Pfaffian. Instead of using Plücker coordinates, they use specific determinants (which, of course, satisfy built-in Plücker relations).…”

“…In [18] and [5], one finds many evaluations of Pfaffians and determinants, with entries which are specializations of Plücker indeterminates. For example, Okada [18,Th.3.4] takes…”

“…Ishikawa [4], Okada [18], M. Ishikawa, S. Okada, H. Tagawa and J. Zeng [5] have given different generalizations of Sundquist's Pfaffian. We show how to connect their results to Th.6 and Th.7.…”

Minor summation formula and a proof of Stanley’s open problem

“…Note that certain determinant and Pfaffian identities of this type first appeared in [16], and applied to solve some alternating sign matrices enumerations under certain symmetries stated in [13]. Certain conjectures which intensively generalize the determinant and Pfaffian identities of this type were stated in [18], and a proof of the conjectured determinant and Pfaffian identities was given in [7]. Now we know that various methods may be adopted to prove this type of identity.…”

“…In the first step we utilize the minor summation formula ( [8]) to express the sum of Schur functions as a Pfaffian. In the second step we express the Pfaffian by a determinant using a Cauchy type Pfaffian formula (also see [17], [18] and [7]), and try to simplify it as much as possible. In the process of this step, it is conceivable that the determinants we treat may be closely related to characters of representations of SP2n and SOm (See [4], [6] and [9]).…”