The Binomial Theorem and motivic classes of universal quasi-split tori

TL;DR

This paper explores the algebraic structure of symmetric powers of varieties, deriving a binomial formula and explicitly describing the classes of universal quasi-split tori within the equivariant Grothendieck group, advancing understanding of motivic classes.

Contribution

It introduces a binomial formula for symmetric powers of varieties and provides explicit expressions for classes of universal quasi-split tori in the equivariant Grothendieck group.

Findings

Derived a binomial formula for symmetric powers of varieties.

Explicitly expressed classes of universal quasi-split tori.

Connected symmetric powers with motivic classes in the Grothendieck group.

Abstract

Taking symmetric powers of varieties can be seen as a functor from the category of varieties to the category of varieties with an action by the symmetric group. We study a corresponding map between the Grothendieck groups of these categories. In particular, we derive a binomial formula and use it to give explicit expressions for the classes of universal quasi-split tori in the equivariant Grothendieck group of varieties.