Adjoint Characters and Odd/Even Characters #
This file formalizes the Hopf-algebraic structure of the adjoint of a B-series method, following Shmelev, Ebrahimi-Fard, Tapia & Salvi, Explicit and Effectively Symmetric Runge-Kutta Methods (arXiv:2507.21006), Sections 4-5.
The canonical involution of the forest algebra sends a forest τ to
(-1)^{|τ|} τ. The adjoint of a B-series character ψ is
ψ*(τ) = (-1)^{|τ|} ψ(S τ), which is the convolution inverse of the involuted
character (Theorem 4.2 of the paper). A B-series method is symmetric exactly
when its character is odd in the sense of Aguiar-Bergeron-Sottile, while
even characters correspond to antisymmetric methods (Corollary 4.3).
Main definitions #
ForestAlgebra.gradingInvolution- the involutionτ ↦ (-1)^{|τ|} τForestAlgebra.LinearFunctional.compGradingInvolution- composition with the involution, which intertwines convolutionForestAlgebra.Character.involution- the induced involution of charactersForestAlgebra.Character.adjoint- the adjoint of a B-series characterForestAlgebra.Character.IsEven,ForestAlgebra.Character.IsOddForestAlgebra.Character.isOdd_iff_adjoint_eq- a method is symmetric, i.e. equal to its adjoint, if and only if its character is odd
The canonical grading involution τ ↦ (-1)^{|τ|} τ of the forest algebra
(arXiv:2507.21006, Definition 4.1).
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Compose a linear functional with the grading involution.
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Composition with the grading involution intertwines convolution, since every term of the BCK coproduct preserves the total order grading.
The canonical involution of characters: ζ̄(τ) = (-1)^{|τ|} ζ(τ)
(arXiv:2507.21006, Section 4).
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The involution of characters is multiplicative with respect to
convolution: (ζξ)‾ = ζ̄ ξ̄ (arXiv:2507.21006, Section 4).
The adjoint of a B-series character ψ, as a linear functional:
ψ*(τ) = (-1)^{|τ|} ψ(S τ) (arXiv:2507.21006, Theorem 4.2). The adjoint
B-series method Ψ* has elementary weights ψ*.
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The adjoint character evaluates as ψ*(τ) = (-1)^{|τ|} ψ(S τ).
The adjoint is a left convolution inverse of the involuted character:
ψ* ψ̄ = e (arXiv:2507.21006, proof of Theorem 4.2).
The adjoint is a right convolution inverse of the involuted character:
ψ̄ ψ* = e (arXiv:2507.21006, proof of Theorem 4.2).
A character is even if it is fixed by the canonical involution. Even characters correspond to antisymmetric methods (arXiv:2507.21006, §5).
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A character is odd if its involution equals its convolution inverse
ζ ∘ S. Odd characters correspond to symmetric B-series methods
(arXiv:2507.21006, Corollary 4.3).
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A B-series method equals its adjoint if and only if its character is odd: symmetric B-series methods are exactly the odd characters (arXiv:2507.21006, Corollary 4.3).
Convolution inverses of linear functionals are unique.
The adjoint is the unique convolution inverse of the involuted character.
The adjoint reverses composition: (Ψ Φ)* = Φ* Ψ*
(the B-series analogue of Hairer-Norsett-Wanner II.8, adjoint of a
composition).
Even characters are closed under convolution: antisymmetric methods form a subgroup of the Butcher group (arXiv:2507.21006, Section 5).
The adjoint of an even (antisymmetric) character is its convolution
inverse ζ ∘ S.
The adjoint of an odd (symmetric) character is the character itself.
The canonical symmetric composition: for any character ψ, the composition
Ψ* ∘ Ψ of a method with its adjoint is symmetric. In group form, the
involution of ψ ψ* is its convolution inverse
(arXiv:2507.21006, Section 5).
The adjoint of a B-series character as a bona fide character, using that the BCK antipode is an algebra morphism.
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ψ*(τ) = (-1)^{|τ|} ψ(S τ) at the character level
(arXiv:2507.21006, Theorem 4.2).
The adjoint character is a right inverse of the involuted character in the character group.
The adjoint character is a left inverse of the involuted character in the character group.
The convolution inverse of a character, as a character: χ ∘ S, using
that the BCK antipode is an algebra morphism.
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The adjoint character is the involution of the inverse character.
Convolution inverses of characters are unique.
Inversion of characters reverses convolution.
Inversion of characters is involutive.
Inversion commutes with the canonical involution.
The involution of the adjoint character is the inverse character.
The inverse of the adjoint character is the involution.
The adjoint is an involution on B-series methods: ψ** = ψ
(arXiv:2507.21006, Section 4).
The canonical symmetric composition ψ* ψ is an odd character
(arXiv:2507.21006, Section 5).
A B-series method is symmetric iff its character equals its adjoint character (arXiv:2507.21006, Corollary 4.3, character form).