Immanant

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 $\operatorname {sgn}$, of S_n, 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 (1934). "Group characters and algebras". Philosophical Transactions of the Royal Society A. 233 (721–730): 99–124. doi:10.1098/rsta.1934.0015. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• 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