The Magic and Mystery of Component Tableaux

Let G be a simple algebraic group over the complex field C, P a parabolic subgroup containing B its Borel subgroup, $P'$ its derived group and m the Lie algebra of its nilradical. The nilfibre N for this action is the zero locus of the augmentation I₊ of the semi-invariant algebra I= C[ m]P'. For $G=SL(n)$ practically nothing was known previously. The only result of comparable, but lesser complexity, is for V:= O∩ n, with O a nilptent G orbit and n the set of strictly upper triangular matrices. Then V is equidimensional with components known as orbital varieties, parameterised by standard tableaux whose shape is dictated by O. Here the components of N are studied for $G=SL(n)$. They increase exponentially in n with no a priori discernable pattern. For each choice of numerical data C, a semi-standard tableau T^ C, is constructed from T. A delicate and tightly interlocking analysis constructs a set of excluded root vectors from m such that the complementary space u^ C has the following properties. First it is a subalgebra of m. Secondly C:=B. u^ C̄ lies in N to which, thirdly, a Weierstrass section can be associated. Fourthly C = dim m-g, where g is the number of generators of the polynomial algebra I. Fifthly the Weierstrass section, is shown to imply that C an irreducible component of N, yet C is only sometimes an orbital variety closure. The resulting Component Map T^ C↦ C is shown to be injective. Evidence for its surjectivity is given.