Immanant of a matrix
- Immanant redirects here; it should not be confused with the philosophical immanent.
Let be a partition of and let be the corresponding irreducible representation-theoretic character of the symmetric group . The immanant of an matrix associated with the character is defined as the expression
The permanent is the case where is the trivial character, which is identically equal to 1.
For example, for matrices, there are three irreducible representations of , as shown in the character table:
As stated above, produces the permanent and produces the determinant, but produces the operation that maps as follows:
- D.E. Littlewood; A.R. Richardson (1934). "Group characters and algebras". Philosophical Transactions of the Royal Society 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.