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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