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On Measuring Singularities of Schubert Varieties in Classical Types

Jeon, Minyoung
http://rave.ohiolink.edu/etdc/view?acc_num=osu1650554365377222

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
This dissertation investigates singularities of Schubert varieties through the study of Hilbert-Samuel mulitiplicities, Kazhdan-Lusztig polynomials and Chern-Mather classes. In Chapter 2, with D. Anderson, T. Ikeda, and R. Kawago, we present local isomorphisms of Schubert varieties associated to vexillary elements in the Weyl group and Schubert varieties in (ordinary, Lagrangian or maximal orthogonal) Grassmannians. We use direct-sum embeddings of flag varieties and local isomorphisms between vexillary Schubert varieties in flag varieties and Schubert varieties in Grassmannians. As an application of the local isomorphisms, we present a combinatorial formula for the multiplicity for vexillary Schubert varieties in flag varieties of types A,B,C and D, in Chapter 3. This extends results in Grassmannians and gives an alternative proof for type A by Li and Yong. This is joint work with Anderson, Ikeda, and Kawago. In Chapter 4, the author establishes algorithmic and inductive formulas for Kazhdan-Lusztig polynomials associated to a covexillary Schubert variety in flag varieties, as another application of the local isomorphisms. The intersection cohomology theory will be used to prove the formulas, relating coefficients associated to a Schubert variety in a flag variety to the one in a Grassmannian. This gives an efficient way of determining the Kazhdan-Lusztig polynomials which are hard to determine by definition. In Chapter 5, the computation for the Chern-Mather class of Schubert varieties in orthogonal (type B or type D) Grassmannians will be given, as an analogous result in ordinary (type A) Grassmannians. Using small resolutions and the equivariant localization, we express the Mather class as a linear combination of classes of sub-Schubert varieties. The Pfaffian formulas arise from this technique, as an analogy of known determinantal formulas for type A.
David Anderson (Advisor)
Roy Joshua (Committee Member)
Hsian-Hua Tseng (Committee Member)
Alexandra Landsman (Committee Member)
127 p.
Mathematics

Recommended Citations

Citations

  • Jeon, M. (2022). On Measuring Singularities of Schubert Varieties in Classical Types [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1650554365377222

    APA Style (7th edition)

  • Jeon, Minyoung. On Measuring Singularities of Schubert Varieties in Classical Types. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1650554365377222.

    MLA Style (8th edition)

  • Jeon, Minyoung. "On Measuring Singularities of Schubert Varieties in Classical Types." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1650554365377222

    Chicago Manual of Style (17th edition)

osu1650554365377222
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