Root pairings taking values in a subring #
This file lays out the basic theory of root pairings over a commutative ring R
, where R
is an
S
-algebra, and the the pairing between roots and coroots takes values in S
. The main application
of this theory is the theory of crystallographic root systems, where S = ℤ
.
Main definitions: #
RootPairing.IsValuedIn
: Given a commutative ringS
and anS
-algebraR
, a root pairing overR
is valued inS
if all root-coroot pairings lie in the image ofalgebraMap S R
.RootPairing.IsCrystallographic
: A root pairing is said to be crystallographic if the pairing between a root and coroot is always an integer.RootPairing.pairingIn
: TheS
-valued pairing between roots and coroots.RootPairing.coxeterWeightIn
: The product ofpairingIn i j
andpairingIn j i
.
If R
is an S
-algebra, a root pairing over R
is said to be valued in S
if the pairing
between a root and coroot always belongs to S
.
Of particular interest is the case S = ℤ
. See RootPairing.IsCrystallographic
.
- exists_value (i j : ι) : ∃ (s : S), (algebraMap S R) s = P.pairing i j
Instances
Alias of RootPairing.IsValuedIn.exists_value
.
A root pairing is said to be crystallographic if the pairing between a root and coroot is always an integer.
Equations
Instances For
A variant of RootPairing.pairing
for root pairings which are valued in a smaller set of
coefficients.
Note that it is uniquely-defined only when the map S → R
is injective, i.e., when we have
[FaithfulSMul S R]
.
Instances For
The S
-span of roots.
Equations
- P.rootSpan S = Submodule.span S (Set.range ⇑P.root)
Instances For
The S
-span of coroots.
Equations
- P.corootSpan S = Submodule.span S (Set.range ⇑P.coroot)
Instances For
A root, seen as an element of the span of roots.
Equations
- P.rootSpanMem S i = ⟨P.root i, ⋯⟩
Instances For
A coroot, seen as an element of the span of coroots.
Equations
- P.corootSpanMem S i = ⟨P.coroot i, ⋯⟩
Instances For
The S
-linear map on the span of coroots given by evaluating at a root.
Equations
- P.root'In S i = (P.corootSpan S).subtype.restrictScalarsRange (Algebra.linearMap S R) ⋯ (P.root' i) ⋯
Instances For
The S
-linear map on the span of roots given by evaluating at a coroot.
Instances For
A variant of RootPairing.coxeterWeight
for root pairings which are valued in a smaller set of
coefficients.
Note that it is uniquely-defined only when the map S → R
is injective, i.e., when we have
[FaithfulSMul S R]
.
Equations
- P.coxeterWeightIn S i j = P.pairingIn S i j * P.pairingIn S j i