Moments of the characteristic polynomial in the three ensembles of random matrices
Moments of the characteristic polynomial of a random matrix taken from any of the three ensembles, orthogonal, unitary or symplectic, are given either as a determinant or a pfaffian or as a sum of determinants. For gaussian ensembles comparing the two expressions of the same moment one gets two remarkable identities, one between an $n\times n$ determinant and an $m\times m$ determinant and another between the pfaffian of a $2n\times 2n$ anti-symmetric matrix and a sum of $m\times m$ determinants.
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