Schubert polynomials and classes of Hessenberg varieties

By David Anderson, Julianna Tymoczko. 2007

49 citations recorded by the source corpus.

Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and we give closed combinatorial formulas for the coefficients in many cases. We introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use our results to determine when such schemes are reduced.

View Paper: https://www.semanticscholar.org/paper/5931b0f782123e47fbc82e7bade396f89f381c91

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Citations in the corpus

• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A (2015, background): “Both of the above types of Hessenberg varieties have been much studied, and it is known, for example, that Hess(N, h) is irreducible [4] and possibly singular [31, 28], while Hess(S, h) is smooth, and possibly non-connected [11].”