Hellenica World

# .

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.