The immanant 
of an 
matrix 
, indexed by a partition 
of 
,
is the weighted sum
 |
where 
is the irreducible group
character of the symmetric group 
indexed by 
(Littlewood and Richardson 1934a). The partitions 
and 
give the permanent and determinant, respectively.
For 
and 
,
the dimension of the corresponding irreducible
representation is 
,
where 
is the identity
permutation. Skandera (2026, p. 16) proved that for 
every 
totally nonnegative
matrix satisfies
 |
The hypothesis requires all minors to be nonnegative, rather than only the entries. The inequalities above concern the specified partitions.
The monomial immanant uses a virtual character dual to the characters induced from trivial representations of Young subgroups. McNulty
(2026) verifies Haiman's positivity, symmetry, and unimodality conjecture for the associated Hecke algebra character polynomials
for all 
. Consequently, the monomial
immanants in this range are nonnegative on totally nonnegative matrices. The exact
computations were performed and cross-checked with Claude Code assistance.
: Kazhdan-Lusztig Positivity of Monomial Immanants."
30 Oct 2026.