Queen Victoria charmed by Carroll’s novel, let it be known to the author that she would be pleased to receive a copy of his forthcoming work. Lewis Carroll was rumoured to have sent her his next volume having the enticing title: “An elementary treatise on determinants”. Thereby Queen Victoria could have acquired practically all necessary to understand the work presented here.
In this lecture we describe the components of an exponentially increasing family of varieties coming from invariant theory.
In more detail, let B be the subgroup of upper triangular matrices in GL(n, ℂ). Then a parabolic subgroup P containing B is given by a composition (c₁, c₂, …, cₖ) of n whose entries are the sizes of Levi factors of P. These are formed from the c_i × c_i blocks down the diagonal, whilst the Lie algebra 𝔪 of the nilradical of P is the subspace lying strictly above these blocks.
Then the variety whose components we seek is the nilfibre N for the adjoint action of P on 𝔪, for all compositions of n and all n. It is the zero locus of the Benlolo–Sanderson generators of the semi-invariant ring ℂ[𝔪]^P′, which are g in number.
We define [4] “The Red Set” which can be viewed as “connected” subsets of the diagonal entries in each Levi block for which just the lowest entry may have multiplicity > 1.
We construct [2] a “component map” from the Red Set to the set of irreducible components of P and show it to be injective. For this one only needs to know how to multiply out a determinant and some very modest intersection theory in complex projective space.
Successive linearisation of invariants gives the components obtained from the component map, but may eliminate some potential ones, since it involves inhomogeneous substitution.
Recently Fittouhi [1] has switched the emphasis to the factorisation coming from equal height columns in a reverse tableau [3]. For a component of dimension ≤ dim V − g, descent on Krull dimension shows that this component lies in one coming from the image of the component map. Alternatively one may recognize that this factorisation only requires excluded root vectors. These involve only homogeneous substitution, so no potential components are eliminated.
