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# Immanant of a matrix

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\times n matrix \( A=(a_{ij}) \) associated with the character \( \chi_\lambda \) is defined as the expression

\( {\rm Imm}_\lambda(A)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)a_{1\sigma(1)}a_{2\sigma(2)}\cdots a_{n\sigma(n)}. \)

The determinant is a special case of the immanant, where \( \chi_\lambda \) is the alternating character \( \sgn \), of Sn, defined by the parity of a permutation.

The permanent is the case where \( \chi_\lambda \) is 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.

References

D.E. Littlewood; A.R. Richardson (1934). "Group characters and algebras". Philosophical Transactions of the Royal Society, Ser. A 233 (721–730): 99–124. doi:10.1098/rsta.1934.0015.

D.E. Littlewood (1950). The Theory of Group Characters and Matrix Representations of Groups (2nd ed.). Oxford Univ. Press (reprinted by AMS, 2006). p. 81.

External links

Immanent at PlanetMath

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