Symmetric ideals of the infinite polynomial ring

Abstract

Let R = C[ξ 1 , ξ 2 , . . .] be the infinite variable polynomial ring, equipped with the natural action of the infinite symmetric group S. We classify the S-primes of R, determine the containments among these ideals, and describe the equivariant spectrum of R. We emphasize that S-prime ideals need not be radical, which is a primary source of difficulty. Our results yield a classification of S-ideals of R up to copotency. Our work is motivated by the interest and applications of S-ideals seen in recent years.

Equivariant algebraic and semi-algebraic geometry of infinite affine space

“…This highlights the special role played by the n = 1 case which we also observe in Section 4 in the semi-algebraic setup. While in the case n = 1 our results are not as strong as the results from [13,14] where a characterization of invariant radical ideals [13] and invariant ideals [14] is given, our results also apply to larger n > 1. Our results lead naturally to Conjecture 3.24 which states that the Kolmogorov quotient of the orbit space is a spectral space, i.e.…”

The spectrum of a twisted commutative algebra

“…See Section 2 for background on Schur functors.) In joint work with Rohit Nagpal [18,19], we study the 𝔖 ∞ -primes in the polynomial ring 𝐂[𝑥 1 , 𝑥 2 , …] and manage to completely classify them. (Here 𝔖 ∞ denotes the infinite symmetric group.)…”

GL-equivariance in positive characteristic I: The infinite exterior algebra

“…The infinite symmetric group. Nagpal-Snowden [NS20,NS21] have classified the S ∞stable ideals in the infinite polynomial ring. There are many remarkable results and open problems [LNNR20, LNNR21, Mur20] about these ideals and their resolutions.…”