Representation stability describes a sequence of representations whose irreducible multiplicities eventually stop changing after the irreducible
labels are adjusted with the size of the group. For a sequence of 
-modules
![V_n= direct sum _lambda(S^(lambda[n]))^( direct sum c_(lambda,n)),](/images/equations/RepresentationStability/NumberedEquation1.svg) |
multiplicity stability means that for every partition 
,
the coefficient 
is independent of 
for all sufficiently large 
. Uniform multiplicity stability
requires a single stable range that works for all 
.
With compatible 
-equivariant maps 
, uniform representation stability additionally
requires these maps to become injective and their 
-orbits to span 
(Church et al. 2015). Borisov (2026) proves that
stabilization of Schur coefficients in the corresponding
Frobenius images is equivalent, subject to finite weight,
to stabilization of coefficients in the monomial
basis of symmetric functions,
and gives bounds for stable ranges.
