• Corpus ID: 235253867

Comparing two perfect bases

@inproceedings{Dranowski2020ComparingTP,
  title={Comparing two perfect bases},
  author={Anne Dranowski},
  year={2020}
}

• Anne Dranowski
• Published 30 May 2021
• Mathematics

Comparing two perfect bases Anne Dranowski Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2020 We study a class of varieties which generalize the classical orbital varieties of Joseph. We show that our generalized orbital varieties are the irreducible components of a Mirković–Vybornov slice to a nilpotent orbit, and can be labeled by semistandard Young tableaux. Furthermore, we prove that Mirković–Vilonen cycles are obtained by applying the Mirković– Vybornov isomorphism to generalized orbital varieties and taking a projective closure, refining Mirković and Vybornov’s result. As a consequence, we are able to use the Lusztig datum of a Mirković–Vilonen cycle to determine the tableau labeling the generalized orbital variety which maps to it, and, hence, the ideal of the generalized orbital variety itself. By homogenizing we obtain equations for the cycle we started with, which is useful for computing various equivariant invariants such as equivariant multiplicity and multidegree. As an application, we show that in type A5 the Mirković–Vilonen basis and Lusztig’s dual semicanonical basis are not equal as perfect bases of the coordinate ring of the unipotent subgroup. This is significant because it shows that perfect bases are not unique. Our comparison relies heavily on the theory of measures developed in [BKK19], so we include what we need. Finally, we state a conjectural combinatorial “formula” for the ideal of a generalized orbital variety in terms of its tableau.