MacMahon's Master Theorem, Representation Theory, and Moments of Wishart Distributions

Abstract

) independently derived a generalization of MacMahon's master theorem. In this article we apply their result to obtain an explicit formula for the moments of arbitrary polynomials in the entries of X, a real random matrix having a Wishart distribution. In the case of the complex Wishart distributions, the same method is applicable. Furthermore, we apply the representation theory of GL d , the complex general linear group, to derive explicit formulas for the expectation of Kronecker products of any complex Wishart random matrix.  2001 Elsevier Science

Algebraic factor analysis: tetrads, pentads and beyond

“…Just as in Example 24, it is impossible to write this invariant as a sum of monomials, but it is very easy to evaluate it numerically using the determinantal representation (21).…”

“…For instance, the derivation of the pentad constraint can be interpreted as an evaluation of the first multilinear resultant (20). Likewise, the second multilinear resultant (21) can be used to produce non-trivial invariants in I p,m when p ≥ 2, m ≥ 8.…”

“…The invariant of degree 24 is obtained evaluating the formula (21), in which we abbreviate the coefficients of f i by a i jk = a i jk (ψ). Just as in Example 24, it is impossible to write this invariant as a sum of monomials, but it is very easy to evaluate it numerically using the determinantal representation (21).…”

Compound Real Wishart and q-Wishart Matrices

“…Explicit formulas for moments of polynomials in one real Wishart matrix appear in [17,18,20,28]. The formulas are more complicated than the formulas for the complex case [16,20,33], and often involve sophisticated tools, like wreath products, Jack polynomials, and properties of hyperoctahedral group.…”

The moments of Wishart processes via Itô calculus

“…The computation of moments of centered Wishart matrices and of multi linear functionals of centered Wishart matrices (involving products of traces) has been studied by many authors for a long time (cf., for example, [11], [7], [12], and the references therein) using methods based on the Laplace transform and the representation theory of the symmetric group. The case of noncentered real Wishart matrices is more difficult and has not been studied sufficiently up to now.…”