A CHARACTERIZATION OF ZERO-DIMENSIONAL ANNIHILATORS OF TABLEAUS

In the theory of linear recursive sequences over a field K there is a natural action of the polynomial ring K[x] on sequences (the end-off l eft shift), and every ideal of K[x] is the annihilator with respect to this action of some sequence s. This is a direct consequence of the f act that K[x] is a principal ideal domain. In dimension n greater than or equal to 2 the analogue of a sequence is a tableau, and there is a n analogous action of K[x(1),x(2),...,x(n)] on tableaus. However, a gi ven ideal of K[X] = K[x(1),x(2),...,x(n)] is not necessarily the annih ilator of some tableau. We will give an example in Section 2. Our main result is to characterize an important subset of those ideals which a re annihilators. More specifically, in the theory of linear recursive sequences, an ideal I of K[x] and a finite set of initial conditions ( the initial fill) completely determine a sequence annihilated by I. Th is is a consequence of the fact that K[x]/I is a finite-dimensional K- vector space. In order for a tableau to be completely determined by an ideal I and a finite set of initial conditions, we must have that K[X ]/I is a finite-dimensional K-vector space, i.e., the only primes cont aining I are maximal, i.e., I is a 0-dimensional ideal, In particular, if m is maximal and I is m-primary then I is 0-dimensional. Using met hods of primary decomposition, we will be able to reduce to this case. Following a suggestion of B. Sturmfels, we show that a 0-dimensional ideal is the annihilator of some tableau if and only if it is Gorenste in. (C) 1997 Academic Press.