Stabilization of invariant chains of polynomial ideals

04/22/2014 - 12:15pm
04/22/2014 - 1:10pm
Ann Johnston (University of Southern California)

Consider an increasing chain of polynomial ideals, where the $k$th ideal is contained inside the polynomial ring with indeterminates labeled $x_1, \ldots, x_k$. Such a chain is said to be symmetrization invariant if the $k$th ideal, $I_k$, is closed under the natural action of the symmetric group, $S_k$, on the indeterminate labels. An invariant chain is said to stabilize if there is some $N$ such that $I_k=S_k I_N$, whenever $k > N$. That is, an invariant chain of ideals stabilizes if, up to the action of the symmetric groups, the chain is (in a sense) finite. In this talk, we will see examples of families of polynomial ideals whose stabilization is guaranteed, and we will explore the underlying theory of stabilization of invariant chains of polynomial ideals. We will also see a concrete application of this theory to a problem in algebraic statistics.

Mudd Science Library 126 (Pomona College)