An ideal in a polynomial ring is called symmetric if it is invariant as a set under the action of the symmetric group, which permutes variables. This talk explores two classes of symmetric ideals: those reflecting the typical behavior of a randomly chosen symmetric ideal, and those enriched with strong combinatorial structure arising from the Borel group. In the first part, motivated by the principle that a general member in a family often exhibits desirable properties, we develop a framework for defining and studying general symmetric ideals with a fixed number of generators up to symmetry. This perspective reveals interesting homological stability phenomena.

In the second part, focusing on symmetric ideals generated by monomials, we introduce the notion of symmetric strongly sifted ideals—the symmetrizations of strongly stable ideals from commutative algebra. We present structural, homological, and combinatorial results for the former ideals, highlighting their connections to discrete polymatroids and permutohedral toric varieties. This talk is based on joint work with Megumi Harada and Liana Sega, as well as separate collaborations with Liana Sega and Alessandra Costantini.