Triple intersection formulas for isotropic Grassmannians

Ravikumar, Vijay

doi:10.2140/ant.2015.9.681

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Triple intersection formulas for isotropic Grassmannians

Vijay Ravikumar

Vol. 9 (2015), No. 3, 681–723

DOI: 10.2140/ant.2015.9.681

Abstract

Let $X$ be an isotropic Grassmannian of type $B$ , $C$ , or $D$ . In this paper we calculate $K$ -theoretic Pieri-type triple intersection numbers for $X$ : that is, the sheaf Euler characteristic of the triple intersection of two arbitrary Schubert varieties and a special Schubert variety in general position. We do this by determining explicit equations for the projected Richardson variety corresponding to the two arbitrary Schubert varieties, and show that it is a complete intersection in projective space. The $K$ -theoretic Pieri coefficients are alternating sums of these triple intersection numbers, and we hope they will lead to positive Pieri formulas for isotropic Grassmannians.

Keywords

triple intersection numbers, isotropic Grassmannian, orthogonal Grassmannian, submaximal Grassmannian, Richardson variety, projected Richardson variety, Pieri rule, $K$-theoretic Pieri formula, $K$-theoretic triple intersection

Mathematical Subject Classification 2010

Primary: 14N15

Secondary: 19E08, 14M15

Milestones

Received: 25 June 2014

Revised: 7 August 2014

Accepted: 7 March 2015

Published: 17 April 2015

Authors

Vijay Ravikumar
Mathematics Department Chennai Mathematical Institute H1, SIPCOT IT Park Siruseri Kelambakkam 603103 India

Vol. 9, No. 3, 2015