Comparing G-sets and quadratic forms

Ricardo Rojas-Echenique has a new paper up on the arXiv comparing $G$-sets and quadratic forms via Dress’s Burnside-to-Grothendieck-Witt map:

• Injectivity and surjectvity of the Dress map

Ricardo was one of my students in the K-group over the summer, and I’m very excited to see this research come to fruition.

Update (May 31, 2016): Ricardo’s paper is now published in JPAA!

The basic idea is as follows: if $L/k$ is a finite Galois extension of fields of characteristic not 2 with Galois group $G$, then there is a natural way to take $G$-orbits to quadratic forms. One assigns to $G/H$ the so-called trace form of the field extension $L^H/k$. This trace form takes $x$ in $L^H$ to ${\mathrm{tr}}_{L^H/k}(x^2)$. If you’re familiar with the Burnside ring $A(G)$ and Grothendieck-Witt ring $GW(k)$, it’s a very fun exercise to work out why this assignment induces a ring homomorphism $h_{L/k}:A(G)\to GW(k)$.

The study of analogies between the Burnside and Grothendieck-Witt rings goes back to Andreas Dress in Appendix B to his 1971 Bielefeld notes. These notes are amazing, and also shockingly difficult to track down. My collaborator Jeremiah Heller found them on microfiche(!) at the UIUC library and kindly scanned them to pdf. If you’d like to take a look yourself, here they are:

• A relation between Burnside- and Wittrings, Appendix B to Notes on the theory of representations of finite groups by A.W.M. Dress, Bielefeld, 1971.

For the uninitiated, allow me to at least briefly expound on the construction of these rings and why they’re so awesome. In the case of the Burnside ring of a finite group $G$, one starts with the set of finite $G$-sets (up to equivariant isomorphism) along with the disjoint union and cartesian product operations. These form a semi-ring, which we can complete to a ring via the Grothendieck construction. This is the Burnside ring $A(G)$. As an abelian group, it is freely generated by orbits $G/H$ where $H$ runs through a set of conjugacy classes of subgroups of $G$.

For the Grothendieck-Witt ring of a field $k$ of characteristic different from 2, you start with quadratic forms (up to isometry) along with direct sum and tensor product operations. Again, you have a semi-ring, and the Grothendieck construction results in the Grothendieck-Witt ring $GW(k)$. As an abelian group, it is generated (but certainly not freely generated) by the one-dimensional quadratic forms $⟨a⟩$ which take $x$ in $k$ to $a{x}^2$. (Here $a$ is a unit in $k$.)

In addition to encoding fascinating representational and arithmetic information about groups and fields, respectively, these rings are also endomorphisms of unit objects in some fashionable stable homotopy categories. The Burnside ring is isomorphic to the endomorphisms of the $G$-equivariant sphere spectrum, while the Grothendieck-Witt ring captures the endomorphisms of the motivic sphere spectrum over $\mathrm{Spec}(k)$.

A number of authors have studied links between $G$-equivariant and motivic stable homotopy theory, including myself and Jeremiah Heller in a paper that was recently accepted to the Transactions of the AMS, Galois equivariance and stable motivic homotopy theory, previously written about on this blog here. We show that there is a functor ${\mathrm{SH}}^G\to {\mathrm{SH}}^k$ from the $G$-equivariant stable homotopy category to the motivic stable homotopy category over $k$ (again where $G=\mathrm{Gal}(L/k)$). Moreover, when $k$ is real closed and $L=k(i)$ (so that $G$ is cyclic of order 2), this functor is full and faithful (at least after completion with respect to the Hopf map). But for all Galois extensions, this functor induces the Dress map $h_{L/k}$ on endomorphisms of the unit objects! Hence the Dress map is the first obstruction to fullness and faithfulness of the functor.

Ricardo’s project, then, was to study when $h_{L/k}$ is either injective or surjective. These are then necessary conditions for faithfulness and fullness of the functor ${\mathrm{SH}}^G\to {\mathrm{SH}}^k$, respectively. More generally, Ricardo’s work tells us something fascinating about when the representation theory of $G$ and quadratic forms over $k$ are nicely related. His result completely classifies injectivity and surjectivity of $h_{L/k}$ in the following two theorems.

Theorem. For a finite nontrivial Galois extension $L/k$, $h_{L/k}$ is injective if and only if $L=k(a^{1/2})$ where $a\in k$ is not a sum of squares.

Theorem. For a finite Galois extension $L/k$, $h_{L/k}$ is surjective if and only if $k$ is quadratically closed in $L$.

The proofs employ elementary but clever arguments, mixing together the expected ingredients: Galois theory, group theory, and the arithmetic of quadratic forms. For more details, I’ll send you to the paper!