The tensor square conjecture asserts that, for each positive integer 
except 2, 4, and 9, some irreducible complex representation of 
has a tensor square containing every irreducible
representation of 
. In partition notation, it
asks for a single 
such that
 |
for every 
.
The partition 
must be self-conjugate, since the multiplicity of the
sign representation is zero otherwise. For example, 
works for 
.
Saxl's conjecture specifies a staircase partition when 
is triangular, a more particular choice than existence of an unspecified partition.
