  This paper focuses on integer linear programs where solutions are binary
matrices, and the corresponding symmetry group is the set of all column
permutations. Orbitopal fixing, as introduced by Kaibel et al., is a technique
designed to break symmetries in the special case of partitioning (resp.
packing) formulations involving matrices with exactly (resp. at most) one
1-entry in each row.The main result of this paper is to extend orbitopal fixing
to the full orbitope, defined as the convex hull of binary matrices with
lexicographically nonincreasing columns.We determine all the variables whose
values are fixed in the intersection of an hypercube face with the full
orbitope.Sub-symmetries arising in a given subset of matrices are also
considered, thus leading to define the full sub-orbitope in the case of the
sub-symmetric group.We propose a linear time orbitopal fixing algorithm
handling both symmetries and sub-symmetries.We introduce a dynamic variant of
this algorithm where the lexicographical order follows the branching decisions
occurring alongthe B&B search. Experimental results for the Unit Commitment
Problem are presented. A comparison with state-of-the-art techniques is
considered to show the effectiveness of the proposed variants of the algorithm.
