A monomial immanant is a matrix function obtained by replacing the irreducible character
in an immanant by a monomial virtual character (Haiman
1993). A virtual character is an integer linear
combination of irreducible group characters.
For a partition 
of 
, let 
be the character induced
from the trivial representation of the
Young subgroup 
of the symmetric group 
. The monomial characters 
are the dual basis
defined by
 |
where 
.
For an 
matrix 
, the monomial immanant is
 |
McNulty (2026) verifies Haiman's positivity conjecture for every 
by exact computation. In particular, all these monomial
immanants are nonnegative on totally
nonnegative matrices. The stronger computation verifies nonnegative coefficients, symmetry,
and unimodality of the associated monomial-character
polynomials on the Kazhdan-Lusztig
basis of the Hecke algebra. This is finite verification,
not a proof for all 
.
McNulty (2026) credits Claude Code with computational work and cross-checks; the
full computation was not independently replayed here.
: Kazhdan-Lusztig Positivity of Monomial Immanants."
30 Oct 2026.