A Case Where Choosing a Product Order Makes the Calculations of a Groebner Basis Much Faster

Let X and Y be generic n by n matrices of indeterminates. Let S = k[x1,…,xr , y1,…, yr] where k is a field of characteristic 0 and r = n2 . Let I S be the ideal generated by the entries of the matrix XY - YX. We will consider the ring S/I and show that it is Cohen-Macaulay for the case n = 4. In order to calculate its Groebner basis we use a product order with 3 blocks of variables and reverse lexicographic order in each block. This makes the computation much smaller and less time consuming.