We describe an explicit geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties so that they break into Schubert varieties. There are no restrictions on the base field, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri’s rule to arbitrary Schubert classes, by way of ex...

The study of Schubert varieties in G/B has led to numerous advances in algebraic combinatorics and algebraic geometry. These varieties are indexed by elements of the corresponding Weyl group, an affine Weyl group, or one of their parabolic quotients. Often times, the goal is to determine which of the algebraic and topological properties of the Schubert variety can be described in terms of the c...

Vanishing and Non-Vanishing Criteria for Branching Schubert Calculus by Kevin Purbhoo Doctor of Philosophy in Mathematics University of California at Berkeley Professor Allen Knutson, Chair We investigate several related vanishing problems in Schubert calculus. First we consider the multiplication problem. For any complex reductive Lie group G, many of the structure constants of the ordinary co...

This thesis presents an expository account of the use of Frobenius splitting techniques in the study of Schubert varieties. After developing the basic theory of Frobenius splitting, we show that the Schubert and Bott-Samelson varieties are split and use this to derive geometric consequences in arbitrary characteristic. The main result highlighted is that Schubert varieties are normal, Cohen-Mac...

— In Section 1 we define and study Schubert unions on l-step flag varieties. This has previously been done in the Grassmann case, i.e., the l = 1 case, by Hansen, Johnsen and Ranestad [7]. In Section 2 we study the algebraic codes given by mapping l-step flag varieties into projective space. This subject has also been treated in the Grassmann case previously (see p. 52). Résumé (Unions de cellu...

Let X = Sp 2n/B the flag variety of the symplectic group. We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of X. We use these polynomials to describe the arithmetic Schubert calculus on X. Moreover, we give a method to compute the natural arithmetic Chern numbers on X, and show that they ar...

We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices.