  While there has been some progress on the decomposition of Kronecker products
of characters of the symmetric groups in recent times, results on the symmetric
and alternating part of Kronecker squares are still scarce. Here, new results
(and conjectures) are presented on this splitting of the squares that
contribute to a refined understanding of the Kronecker squares. Furthermore,
connections to 2-modular decomposition numbers, Catalan combinatorics, and to
the Saxl conjecture are discussed which further motivate the study of these
splittings.
