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Patch ideals and Peterson varieties

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Abstract

Patch ideals encode neighbourhoods of a variety in GL n /B. For Peterson varieties we determine generators for these ideals and show they are complete intersections, and thus Cohen–Macaulay and Gorenstein. Consequently, we

  • — combinatorially describe the singular locus of the Peterson variety;

  • — give an explicit equivariant K-theory localization formula; and

  • — extend some results of [B. Kostant ‘96] and of D. Peterson to intersections of Peterson varieties with Schubert varieties.

We conjecture that the tangent cones are Cohen–Macaulay, and that their h-polynomials are nonnegative and upper-semicontinuous. Similarly, we use patch ideals to briey analyze other examples of torus invariant subvarieties of GL n /B, including Richardson varieties and Springer fibers.

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Authors and Affiliations

  1. Department of Chemistry and Mathematics, Florida Gulf Coast University, Fort Myers, FL, 33965-6565, USA

    Erik Insko

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Alexander Yong

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  1. Erik Insko
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  2. Alexander Yong
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Correspondence to Erik Insko.

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Insko, E., Yong, A. Patch ideals and Peterson varieties. Transformation Groups 17, 1011–1036 (2012). https://doi.org/10.1007/s00031-012-9201-x

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