coxeterWeightIn — Mathlib

English

The dual annihilator map identity holds with the flip map and relates to the span of roots.

Русский

Тождество двойного аннигилирующего отображения сохраняется с flips и связано с охватом корней.

LaTeX

$$$(P.rootSpan R).dualAnnihilator.map P.flip.toPerfPair.symm = span R (range P.root') . dualCoannihilator$$

Lean4

/-- 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]`. -/
def coxeterWeightIn (S : Type*) [CommRing S] [Algebra S R] [P.IsValuedIn S] (i j : ι) : S :=
  P.pairingIn S i j * P.pairingIn S j i

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