Minor summation formula of pfaffians

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

“…Before we proceed to the proof of Theorem 2.1, we cite a lemma from [8]. The proof is not difficult, but we omit the proof and the reader can consult [8], Section 4, Lemma 7.…”

“…denote the set of all r-element subsets of S. We cite a theorem from [8] which we call a minor summation formula: Theorem 2.2. Let n and N be non-negative integers such that 2n ≤ N .…”

“…Our proof proceeds by three steps. 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.…”

Explicit Capelli Identities for Skew Symmetric Matrices

“…Since in [4] the skew Capelli elements are introduced through their eigenvalues in terms of the irreducible summands of the decomposition of the ring of polynomials on the space of skew-symmetric matrices, it is difficult to obtain explicit formulae directly from the definition. The main purpose of the present paper is to provide such an explicit Capelli identity by using the minor summation formula of Pfaffians established in [5]. In particular, we give an explicit formula for the skew Capelli element, which in fact belongs to the centre ZU(gl N ) of the universal enveloping algebra U(gl N ), in terms of the trace of powers of a matrix E = E N .…”

“…To find the skew Capelli elements, we look at the following identity as our starting point (see p. 296, Remark 2 in [5]):…”