Abstract
PROPERTIES OF SINGULAR SCHUBERT VARIETIES SEPTEMBER 2013 JENNIFER KOONZ, B.A., WELLESLEY COLLEGE Certificate, SMITH COLLEGE, CENTER FOR WOMEN IN MATHEMATICS M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor ERIC SOMMERS This thesis deals with the study of Schubert varieties, which are subsets of flag varieties indexed by elements of Weyl groups. We start by defining Lascoux elements in the Hecke algebra, and showing that they coincide with the Kazhdan-Lusztig basis elements in certain cases. We then construct a resolution (Zw, π) of the Schubert variety Xw for which Rπ∗(C[`(w)]) is a sheaf on Xw whose expression in the Hecke algebra is closely related to the Lascoux element. We also define two new polynomials which coincide with the intersection cohomology Poincare polynomial in certain cases. In the final chapter, we discuss some interesting combinatorial results concerning Bell and Catalan numbers which arose throughout the course of this work.
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