Immanant of a matrix

Immanant of a matrix

:"Immanant redirects here, it should not be confused with the philosophical immanent."

In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent.

Let lambda=(lambda_1,lambda_2,ldots) be a partition of n and let chi_lambda be the corresponding irreducible representation-theoretic character of the symmetric group S_n. The "immanant" of an n imes n matrix A=(a_{ij}) associated with the character chi_lambda is defined as the expression

:{ m Imm}_lambda(A)=sum_{sigmain S_n}chi_lambda(sigma)a_{1sigma(1)}a_{2sigma(2)}cdots a_{nsigma(n)}.

This generalizes the notions of determinant and permanent, the determinant being the case where chi_lambda is the alternating character sgn, and the permanent being the case where chi_lambda the trivial character, which is identically equal to 1.Littlewood and Richardson also studied its relation to Schur functions in the representation theory of the symmetric group.

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