Hermitian K-theory of Grassmannians

TL;DR

This paper computes the Hermitian K-theory of even-dimensional Grassmannians over characteristic zero fields, revealing a structure composed of base field K-theory and GW-theory components linked to Young diagram symmetries.

Contribution

It introduces a method to determine the Hermitian K-theory of Grassmannians using symmetries of Young diagrams, connecting geometric structures to algebraic K-theory.

Findings

Decomposition of Hermitian K-theory into sums of base field K-theory and GW-theory.

Identification of symmetric and asymmetric Young diagrams as indexing sets.

Explicit computation over fields of characteristic zero.

Abstract

We compute the additive structure of the Hermitian $K$-theory spectrum of an even-dimensional Grassmannian over a base field $k$ of characteristic zero in terms of the Hermitian $K$-theory of $X$, using certain symmetries on Young diagrams. The result is a direct sum of copies of the $K$-theory of the base field and copies of the $GW$-theory of the base field, indexed by \emph{asymmetric} and \emph{symmetric} Young diagrams, respectively.