A survey chapter presenting the cohomology of flag varieties and Schubert and Schur polynomials as the rigorous basis for solving Schubert's enumerative geometry problems.
On the Likelihood of Comparability in Bruhat Order
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abstract
The poset of permutations of [n] under Bruhat ordering is studied. We give nontrivial upper and lower bounds for the number of comparable pairs of permutations in both the weak and strong versions of this order. In light of numerical experiments, we conjecture that in either case the upper bound is qualitatively close to the actual number of comparable pairs.
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Introduction to the Cohomology of the Flag Variety
A survey chapter presenting the cohomology of flag varieties and Schubert and Schur polynomials as the rigorous basis for solving Schubert's enumerative geometry problems.