Markus Hunziker

Kostant Modules in Blocks of Category 𝒪 S

In this article, the authors investigate infinite-dimensional representations L in blocks of the relative (parabolic) category 𝒪 S for a complex simple Lie algebra, having the property that the cohomology of the nilradical with coefficients in L “looks like” the cohomology with coefficients in a finite-dimensional module, as in Kostants theorem. A complete classification of these “Kostant modules” in regular blocks for maximal parabolics in the simply laced types is given. A complete classification is also given in arbitrary (singular) blocks for Hermitian symmetric categories.

FREE FERMIONIC HETEROTIC MODEL BUILDING AND ROOT SYSTEMS

We consider an alternative derivation of the GSO Projection in the free fermionic construction of the weakly coupled heterotic string in terms of root systems, as well as the interpretation of the GSO Projection in this picture. We then present an algorithm to systematically and efficiently generate input sets (i.e. basic vectors) in order to study landscape statistics with minimal computational cost. For example, the improvement at order 6 is ≈10-13 over a traditional brute force approach, and improvement increases with order. We then consider an example of statistics on a relatively simple class of models.

Diagrams of Hermitian type, highest weight modules, and syzygies of determinantal varieties

In this mostly expository paper, a natural generalization of Young diagrams for Hermitian symmetric spaces is used to give a concrete and uniform approach to a wide variety of interconnected topics including posets of noncompact roots, canonical reduced expressions, rational smoothness of Schubert varieties, parabolic Kazhdan–Lusztig polynomials, equivalences of categories of highest weight modules, BGG resolutions of unitary highest weight modules, and finally, syzygies and Hilbert series of determinantal varieties.

Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood--Richardson Coefficients

Let K be a complex reductive algebraic group and V a representation of K . Let S denote the ring of polynomials on V . Assume that the action of K on S is multiplicity-free. If λ denotes the isomorphism class of an irreducible representation of K , let ρλ : K → GL(Vλ) denote the corresponding irreducible representation and Sλ the λ-isotypic component of S. Write Sλ · Sμ for the subspace of S spanned by products of Sλ and Sμ. If Vν occurs as an irreducible constituent of Vλ ⊗ Vμ, is it true that Sν ⊆ Sλ · Sμ? In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring S to the usual Littlewood–Richardson rule. Department of Mathematics, The University of Georgia, Athens, GA 30602-7403, USA e-mail: wag@math.uga.edu Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA e-mail: Markus Hunziker@baylor.edu Received by the editors June 7, 2006. This material is based upon work supported by the National Science Foundation under Grant No. 0403838. AMS subject classification: Primary 14L30; secondary 22E46.