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 \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.

For example, for 3 \times 3 matrices, there are three irreducible representations of S_3, as shown in the character table:

S_3 e (1\ 2) (1\ 2\ 3)
\chi_1 1 1 1
\chi_2 1 -1 1
\chi_3 2 0 -1

As stated above, \chi_1 produces the permanent and \chi_2 produces the determinant, but \chi_3 produces the operation that maps as follows:

\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \rightsquigarrow 2 a_{11} a_{22} a_{33} - a_{12} a_{23} a_{31} - a_{13} a_{21} a_{32}

Littlewood and Richardson also studied its relation to Schur functions in the representation theory of the symmetric group.

References[edit]

  • 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[edit]