Column group action

A column group action often means the action of a matrix group on column vectors by left multiplication: $g\cdot v=gv$. The rule $(gh)v=g(hv)$ makes this a left group action, and the identity matrix fixes every column.

The same term is used in Young symmetrizers for the group permuting entries within each column of a tableau. In that setting the action permutes labels rather than multiplying a coordinate column. The surrounding representation must specify which meaning is intended. For a concrete coordinate example, a permutation matrix exchanges two entries of a column vector.