Abstract:
1. Let A n⇥n be a symmetrizable generalized Cartan matrix (GCM) and let g be the associated symmetrizable Kac-Moody Lie algebra with a fixed Cartan subalgebra h. Parabolic Verma modules are highest weight modules for g that simultaneously generalize irreducible integrable modules and Verma modules of g. They are indexed by (, I) where 2 h ⇤ and I ⇢ {1 i n : (↵ _ i ) is a non-negative integer}. In the first part of the thesis we give a necessary and sucient condition for when products of characters of parabolic Verma modules (and their restrictions to some subalgebras of h) are equal. This extends the results of C.S. Rajan [23] andVenkatesh-Viswanath [26] to a class of typically reducible modules.
2. Schur polynomials form a distinguished basis for the ring of symmetric polynomials. The second part of the thesis extends a theorem of Littlewood [16] that asserts that under the action of the map t (which is the adjoint to the map “plethysm by the power sum symmetric function P t ”) the Schur polynomial s factorizes into a product of t many Schur polynomials indexed by the t-quotients of . More precisely, we generalize this fact to a class of flagged skew Schur polynomials. This includes an interesting family of key polynomials as a special case. As an aside we obtain a family of pattern avoiding permutations that are enumerated by the Fuss-Catalan numbers.
