# Zero-cycles with coefficients for the second generalized symplectic involution variety of an algebra of degree 4

## TL;DR

This paper computes the group of $K_1$-zero-cycles on a specific algebraic variety associated with degree 4 algebras with symplectic involution, linking it to multipliers of similitudes.

## Contribution

It provides a new explicit description of $K_1$-zero-cycles for the second generalized involution variety of degree 4 algebras with symplectic involution, using the framework of Chernousov and Merkurjev.

## Key findings

- Explicit computation of $K_1$-zero-cycles for the variety.
- Connection established between zero-cycles and multipliers of similitudes.
- Application to homogeneous varieties of type $ ext{C}_2$.

## Abstract

We compute the group of $K_1$-zero-cycles on the second generalized involution variety for an algebra of degree 4 with symplectic involution. This description is given in terms of the group of multipliers of similitudes associated to the algebra with involution. Our method utilizes the framework of Chernousov and Merkurjev for computing $K_1$-zero-cycles in terms of $R$-equivalence classes of prescribed algebraic groups. This gives a computation of $K_1$-zero-cycles for some homogeneous varieties of type $\mathsf{C}_2$.