Monoid-algebra elements that annihilate a fixed vector

An element a of the monoid algebra k[G] acting through a representation ρ kills a vector v when three conditions meet: doubling is injective on V, some g fixes v, and right multiplication by g negates a. The last two make ρ.asAlgebraHom a v its own negative, and injective doubling turns being its own negative into vanishing.

That is the mechanism behind the column-antisymmetrizer vanishing arguments of TauCeti/RepresentationTheory/Symmetric/, which are its consumers: the antisymmetrizer of a set of indices absorbs each permutation of those indices up to its sign, so against a vector fixed by an odd such permutation the two conditions hold and the action is zero.

Nothing here is specific to symmetric groups or to ℚ. G is a monoid, and the module and the scalars are arbitrary; nothing is asked of 2 in k at all: the hypothesis is that doubling is injective on V, taken as an explicit assumption rather than read off the scalars. So this covers torsion-free modules over ℤ, where 2 is not a unit, and equally modules over a ring with zero divisors whose additive group has no 2-torsion.

Main results

• Representation.asAlgebraHom_eq_zero_of_mul_single_eq_neg: with doubling injective on V, an algebra element negated by right multiplication by an element fixing v annihilates v.

theorem Representation.asAlgebraHom_eq_zero_of_mul_single_eq_neg {k : Type u_1} {G : Type u_2} {V : Type u_3} [CommRing k] [Monoid G] [AddCommGroup V] [Module k V] (h2inj : Function.Injective fun (w : V) => 2 • w) (ρ : Representation k G V) {a : MonoidAlgebra k G} {g : G} {v : V} (hfix : (ρ g) v = v) (hneg : a * MonoidAlgebra.single g 1 = -a) : (ρ.asAlgebraHom a) v = 0

An algebra element absorbed by a fixing element, up to sign, annihilates the vector. If doubling is injective on V, g fixes v, and right multiplication by single g 1 negates a, then a acts as zero on v.