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This follows from the Jacobi-Trudi identity expressing the Schur functions through elementary symmetric functions, the latter being polynomials in matrix entries when evaluated at the eigenvalues.
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This follows from the Jacobi-Trudi identity expressing the Schur functions through elementary symmetric functions, the latter being polynomials in matrix entries when evaluated at the eigenvalues.
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The structure of Eq. 18 has been verified by symbolical computation of Eq. 17 in Mathematica, however, its exact analytical proof is still lacking. Our analysis suggests that for those partitions which have all the aj 's distinct, the corresponding bj 's are given by bj = aj - λj (2/β-1). In the general case, finding similar relation between these two objects is an interesting open problem.
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Pfaffian, or quaternion determinant, of an antisymmetric matrix A is defined as follows: (Ref.) Pfaff (A) = det (A). It is important to note that a Pfaffian is analytic in the matrix entries. Also, in the particular case of unitary symmetry, β=2, one generally has A=i σ2 □ A, with σ2 being the corresponding Pauli matrix in the quaternion subspace, thus Pfaff (A) ≡det (A) in this case.
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