A Kronecker coefficient 
is the multiplicity of the complex Specht module 
in 
for the symmetric
group 
, where all three partitions
have the same size 
. Equivalently, writing 
for its group character,
 |
(1)
|
It is a nonnegative integer and is symmetric in the three partitions. For example, 
for 
contains the trivial
representation, the sign representation,
and the two-dimensional irreducible representation,
each once. Saxl's conjecture requires positivity
of these coefficients when the first two partitions
are the same staircase.
Let 
denote the dimension of 
. For every fixed 
, Slonim (2026) reports the uniform asymptotic
 |
(2)
|
as 
over all triples satisfying 
.
In particular, all these coefficients are eventually
positive. The dimension condition describes deterministic sequences of partitions,
not a probabilistic assertion about a random partition.
For the staircase 
of size 
and the square 
of size 
, Slonim (2026) reports
 |
(3)
|
and
 |
(4)
|
The dimension deficit of a partition 
means 
. For these
special shapes it is proportional to their sizes, so they are outside the preceding
dimension range. Slonim attributes the mathematical statements and proofs to ChatGPT
6 Pro and states that the technical proofs have not been independently verified.
The same work reports dimension-deficit and vanishing results for Littlewood-Richardson
coefficients.
