  We call one-way simple random walk a random walk in the quadrant Z_+^n whose
increments belong to the canonical base. In relation with representation theory
of Lie algebras and superalgebras, we describe the law of such a random walk
conditioned to stay in a closed octant, a semi-open octant or other types of
semi-groups. The combinatorial representation theory of these algebras allows
us to describe a generalized Pitman transformation which realizes the
conditioning on the set of paths of the walk. We pursue here in a direction
initiated by O'Connell and his coauthors [13,14,2], and also developed in [12].
Our work relies on crystal bases theory and insertion schemes on tableaux
described by Kashiwara and his coauthors in [1] and, very recently, in [5].
