  Supersymmetry and R-symmetry Ward identities relate on-shell amplitudes in a
supersymmetric field theory. We solve these Ward identities for (Next-to)^K MHV
amplitudes of the maximally supersymmetric N=4 and N=8 theories. The resulting
superamplitude is written in a new, manifestly supersymmetric and R-invariant
form: it is expressed as a sum of very simple SUSY and SU(N)_R-invariant
Grassmann polynomials, each multiplied by a "basis amplitude". For (Next-to)^K
MHV n-point superamplitudes the number of basis amplitudes is equal to the
dimension of the irreducible representation of SU(n-4) corresponding to the
rectangular Young diagram with N columns and K rows. The linearly independent
amplitudes in this algebraic basis may still be functionally related by
permutation of momenta. We show how cyclic and reflection symmetries can be
used to obtain a smaller functional basis of color-ordered single-trace
amplitudes in N=4 gauge theory. We also analyze the more significant reduction
that occurs in N=8 supergravity because gravity amplitudes are not ordered. All
results are valid at both tree and loop level.
