Let be a compact oriented
surface of genus with one
boundary component, and
its mapping class group. Morita showed that the image of the
Johnson
homomorphism
of is contained
in the kernel of an
–equivariant surjective
homomorphism ,
where and
is the degree
part of the
free Lie algebra
generated by .
In this paper, we study the –module
structure of the cokernel of the
rational Johnson homomorphism ,
where . In particular, we show
that the irreducible –module
corresponding to a partition
appears in the Johnson
cokernel for any
and with
multiplicity one. We also give a new proof of the fact due to Morita that the irreducible
–module corresponding
to a partition
appears in the Johnson cokernel with multiplicity one for odd
.
The strategy of the paper is to give explicit descriptions of maximal vectors with highest
weight
and in
the Johnson cokernel. Our construction is inspired by the Brauer–Schur–Weyl duality
between
and the Brauer algebras, and our previous work for the Johnson cokernel of the
automorphism group of a free group.