Browsing by Subject "Representation theory."
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Item Global SL(2,R) representations of the Schrödinger equation with time-dependent potentials.(2012-08-08) Franco, Jose A.; Sepanski, Mark R. (Mark Roger); Mathematics.; Baylor University. Dept. of Mathematics.We study the representation theory of the solution space of the one-dimensional Schrödinger equation with time-dependent potentials that possess sl₂-symmetry. We give explicit local intertwining maps to multiplier representations and show that the study of the solution space for potentials of the form V (t, x) = g₂(t)x²+g₁(t)x+g₀ (t) reduces to the study of the potential free case. We also show that the study of the time-dependent potentials of the form V (t, x) = λx⁻² + g₂(t)x² + g₀(t) reduces to the study of the potential V (t, x) = λx⁻². Therefore, we study the representation theory associated to solutions of the Schrödinger equation with this potential only. The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category.Item On quasi-dominant weights and Hilbert series of determinantal varieties.(2014-09-05) Alexander, Jordan, 1984-; Hunziker, Markus, 1968-; Mathematics.; Baylor University. Dept. of Mathematics.The coordinate rings of the classical determinantal varieties are each isomorphic to a classical invariant ring by Weyl's fundamental theorems of invariant theory. Since these rings are Cohen-Macaulay, their Hilbert series are rational functions whose numerator polynomials have nonnegative integer coefficients. In the case of general determinantal varieties, as well as in the case of symmetric determinantal varieties, these numerator polynomials were shown to be equal to the Hilbert series of certain finite-dimensional highest weight modules and were given an explicit combinatorial description. The current work extends these results to the alternating determinantal varieties. The proof of these results, in all three cases, relies on the fact that the coordinate rings of the determinantal varieties carry the structure of a Wallach representation. The Hilbert series of the Wallach representation is a rational function whose numerator polynomial is given by the Hilbert series of a finite-dimensional highest weight module, and the Hilbert series of the determinantal variety is equal to the Hilbert series of the Wallach representation. T. J. Enright and J. F. Willenbring introduced the more general class of quasi-dominant weights and showed that if L is a unitarizable highest weight module with quasi-dominant highest weight, then the Hilbert series of L is of the form H_L(t) = R * (H_E(t)) / ((1-t)^D), where R is a rational number, E is a finite-dimensional highest weight module, and D is the Gelfand-Kirillov dimension of L. The set of quasi-dominant weights has an interesting characterization in terms of parabolic category O and Kostant's minimal length coset representatives. We give a new characterization in terms of associated varieties and show that the subset of quasi-dominant weights whose highest weight modules occur in the setting of Howe dual pairs has a nice description in terms of the highest weights of the "Howe dual'' representations. Finally, we give some new results on the number of quasi-dominant reduction points.