Root Data

A root datum keeps the character and cocharacter lattices separate. The project follows Mathlib's convention: a root datum is an integer-valued root pairing, with the usual finiteness assumptions supplied independently.

Definition

Let $X$ and $X^\vee$ be abelian groups. In Mathlib,

$$ RootDatum\ \iota\ X\ X^\vee := RootPairing\ \iota\ \mathbb Z\ X\ X^\vee. $$

Unpacked, an element $P:RootDatum\ \iota\ X\ X^\vee$ contains:

a perfect bilinear pairing

$$ \langle-,-\rangle:X\times X^\vee\longrightarrow\mathbb Z; $$

injections $i\mapsto\alpha_i$ into $X$ and $i\mapsto\alpha_i^\vee$ into $X^\vee$;
the normalization $\langle\alpha_i,\alpha_i^\vee\rangle=2$;
permutations $r_i$ of the common index type $\iota$ satisfying

$$ \alpha_{r_i(j)} =\alpha_j-\langle\alpha_j,\alpha_i^\vee\rangle\alpha_i, $$

and

$$ \alpha_{r_i(j)}^\vee =\alpha_j^\vee-\langle\alpha_i,\alpha_j^\vee\rangle\alpha_i^\vee. $$

The images $\Phi=\{\alpha_i\}\subseteq X$ and $\Phi^\vee=\{\alpha_i^\vee\}\subseteq X^\vee$ are the roots and coroots. Using a common index type makes the root--coroot bijection and its compatibility with reflections part of the data.

Finite reductive root data

For the root datum of a connected reductive group, one additionally assumes:

$X$ and $X^\vee$ are finitely generated; perfect pairing then gives finite-rank free abelian groups;
the indexing type $\iota$ is finite;
$P.IsReduced$.

These are mixins around $RootDatum$, not fields duplicated inside the abbreviation.

No spanning condition is imposed on $\Phi$ inside $X$. This is essential: the quotient direction records the central torus, and even for a semisimple group the root lattice may have finite index in the character lattice.

Semisimple and toral cases

The datum is semisimple when the roots span after rationalization:

$$ \operatorname{span}_{\mathbb Q}(\Phi) =X\otimes_{\mathbb Z}\mathbb Q. $$

It is toral when $\iota$ is empty. A general reductive root datum may contain both a semisimple part and a central torus part.

One should not express semisimplicity by placing $P.IsRootSystem$ directly on the integer root datum: that would require the roots to span $X$ over $\mathbb Z$, which is stronger than rational spanning and depends on the isogeny form.

Root datum of a reductive group

For a split connected reductive group $G$ with split maximal torus $T$,

$$ X=X^*(T),\qquad X^\vee=X_*(T), $$

with the evaluation pairing. The roots are the nonzero weights of $T$ on $\operatorname{Lie}(G)$, and the coroots are the associated one-parameter subgroups. This construction is treated separately in [[node:root_data.root_datum_from_reductive_group|Root Datum of a Reductive Group]].

Duality

The dual root datum exchanges the two lattices and the two injections:

$$ (X,\Phi,X^\vee,\Phi^\vee)^\vee =(X^\vee,\Phi^\vee,X,\Phi). $$

This is the combinatorial operation underlying Langlands duality.

References

N. Bourbaki, Groupes et algebres de Lie, Chapters IV--VI.
B. Conrad, Reductive Group Schemes, Appendix B.
Mathlib, LinearAlgebra.RootSystem.Defs and LinearAlgebra.RootSystem.Reduced. The isomorphism between root data and split reductive groups (the Chevalley--Tits classification) is not yet an equivalence of categories in Mathlib.