Return to search

The Refined Solution to the Capelli Eigenvalue Problem for gl(mjn)+gl(mjn) and gl(mj2n)

In this thesis, we consider the question of describing the eigenvalues of a distinguished family of invariant differential operators associated to a Lie superalgebra g and a g-module W, called the "Capelli basis", via evaluation of certain classes of supersymmetric functions, called the interpolation super Jack polynomials. Finding the eigenvalues of the Capelli basis is referred to the Capelli Eigenvalue Problem. The eigenvalue formula depends on the chosen parametrization of the highest weight vectors in the decomposition of the superpolynomial algebra P(W), and consequently on the choice of a Borel subalgebra. In this thesis, we give a solution for each conjugacy class of Borel subalgebras, which we call a refined solution to the Capelli Eigenvalue Problem.

Given the pair (g, W), we investigate the formulae for the eigenvalues of the Capelli operators associated to the completely reducible and multiplicity-free modules for two cases: diagonal and symmetric cases. In the former case, we show that we can express the eigenvalue of the Capelli operator on the irreducible component of the multiplicity-free decomposition of P(W) as a polynomial function of the b-highest weight of the irreducible component for any Borel subalgebra b.

In the latter case, we show with a concrete counterexample that we cannot expect the results to be as strong as in the first case for all Borel subalgebras. We then express the eigenvalue of the Capelli operator on the irreducible component of the multiplicity-free decomposition of P(W) as a polynomial function of a piecewise affine map on the span of b-highest weights of the irreducible submodules of P(W), with respect to different decreasing Borel subalgebras b.

Identiferoai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/44422
Date22 December 2022
CreatorsMengyuan, Cao
ContributorsNevins, Monica, Salmasian, Hadi
PublisherUniversité d'Ottawa / University of Ottawa
Source SetsUniversité d’Ottawa
LanguageEnglish
Detected LanguageEnglish
TypeThesis
Formatapplication/pdf
RightsAttribution 4.0 International, http://creativecommons.org/licenses/by/4.0/

Page generated in 0.0023 seconds