Immanant of a matrix

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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 $χ λ$ be the corresponding irreducible representation-theoretic character of the symmetric group $S n$ . The immanant of an $n\times n$ matrix $A = (a i j)$ associated with the character $χ λ$ 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 $χ λ$ is the alternating character $sgn$ , of S_n, defined by the parity of a permutation.

The permanent is the case where $χ λ$ 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.

[edit] References

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

[edit] External links

Immanent at PlanetMath

Immanant of a matrix

From Wikipedia, the free encyclopedia

[edit] References

[edit] External links

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