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Abstract: Given a linear action of a group G on a K-vector space V, we consider the invariant ring , where is the dual space. We are particularly interested in the case where and G is the group of all upper unipotent matrices or the group of all upper triangular matrices in . In fact, we determine for and . The result is a complete intersection for all values of n and q. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list of “doubly parametrized” series of group actions whose invariant rings are known to have a uniform description. [Copyright &y& Elsevier]