A Littlewood-Richardson coefficient 
is the nonnegative
integer coefficient in the expansion of a product
of Schur polynomials:
 |
(1)
|
The partitions satisfy 
when the coefficient
is nonzero. The identity is understood in sufficiently many variables, or in the
ring of symmetric functions.
Equivalently, for 
and 
, 
is the multiplicity of the Specht module 
in the representation of 
induced from 
for 
(Macdonald 1995). Thus these coefficients
concern induction between different symmetric groups,
whereas a Kronecker coefficient concerns
a vector space tensor product of representations
of the same symmetric group. For example, 
.
The Littlewood-Richardson rule counts the semistandard skew tableaux of shape 
and content 
whose row-reading word is a lattice word. Rows are read from
right to left, beginning with the top row. Lattice means that in every initial segment
the number of occurrences of 
is at least the number of occurrences of 
, for every positive
integer 
(Macdonald 1995).
Write 
for the dimension of the
Specht module indexed by a partition 
of size 
, and define its dimension deficit by
 |
(2)
|
For 
and 
, put 
.
Slonim (2026) reports that, for every fixed 
,
there are positive constants 
and 
such that
 |
(3)
|
whenever 
and 
.
The requirement that the coefficient be positive
is essential to this finite-deficit statement.
Large dimensions alone do not force positivity. For every fixed 
, Slonim (2026) reports an 
such that, for every sufficiently large even 
and every pair of partitions 
and 
of 
with 
, there is
a partition 
of 
with 
and 
. This assertion concerns vanishing
coefficients, not an arbitrarily small positive coefficient.
For 
, the same work reports 
. When 
and the diagram of 
is contained in that of 
, equality holds iff the
skew shape 
has at most one box in each row and each column. For
a shape of that kind, equality holds for every partition 
of its size.
Let 
and let 
be the largest 
over partitions
of sizes 
, 
,
and 
. Slonim (2026) reports the exact stabilization threshold
 |
(4)
|
for 
and 
. The exceptions are 
, 
,
and 
, for which 
for every 
,
using the empty partition at 
. The equality construction uses a skew shape consisting
of 
boxes with at most one in each row and each column. For example, at 
with 
,
take 
and 
.
Slonim attributes the mathematical statements and proofs to ChatGPT 6 Pro and explicitly states that the technical proofs have not been independently verified.
