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On the Hermitian K-theory of Grassmannians

Sunny Sood, Chunkai Xu

Abstract

We compute the $\mathbb{G}W$-spectrum (Karoubi--Grothendieck--Witt spectrum) of Grassmannians over divisorial schemes defined over fields of characteristic zero, and, as a corollary, determine their stabilized $\mathbb{L}$-theory spectrum. As part of our computation, we establish split fibration sequences in $\mathbb{K}$-theory, $\mathbb{G}W$-theory and $\mathbb{L}$-theory.

On the Hermitian K-theory of Grassmannians

Abstract

We compute the -spectrum (Karoubi--Grothendieck--Witt spectrum) of Grassmannians over divisorial schemes defined over fields of characteristic zero, and, as a corollary, determine their stabilized -theory spectrum. As part of our computation, we establish split fibration sequences in -theory, -theory and -theory.

Paper Structure

This paper contains 27 sections, 30 theorems, 144 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Some combinatorial notions
  5. Derived category of Grassmannians
  6. A canonical pairing
  7. Computation of a combinatorial number
  8. $\mathbb{G}W$-spectrum of Grassmannians
  9. Computation of the mutation
  10. Push-forward in Hermitian $K$-theory, by M. Schlichting
  11. Introduction
  12. Push forward for codimension $1$ regular embeddings
  13. Push forward for projective bundle projections
  14. Push forward for regular embeddings
  15. Strictly perfect complexes on dg schemes
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $S$ be a divisorial scheme defined over a field of characteristic zero. Let $k \leq n$ and let $\operatorname{Gr}^{k,n}_{S}$ denote the Grassmannian over $S$. Let $\mathcal{Q}$ denote the tautological bundle of rank $k$ on $\operatorname{Gr}^{k,n}_{S}$. Then, where $R(k,l)$ denotes the number of Young diagrams of height at most $k$ and width at most $l$, $S(k,l)$ denotes the number of symmetr

Figures (2)

  • Figure 1: Illustrating two Young diagrams which are complement to each other within a given rectangle.
  • Figure 2: The diagram on the left is $\lambda$, the one on the right is $\lambda^\circ$

Theorems & Definitions (74)

  • Theorem 1.1: See \ref{['thm:compute']} below
  • Theorem 1.2: See Theorem \ref{['thm:split']} below
  • Corollary 1.3: See \ref{['cor:Ksplit']}
  • Corollary 1.4: see \ref{['cor:Ltheory']} below
  • Remark 1.5
  • Remark 1.6
  • Proposition 2.1: kapranov1988derived
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4: Littlewood--Richardson Rule
  • ...and 64 more