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A<sc>bstract</sc> In this paper, we propose a construction of GLSM defects corresponding to Schubert cycles in Lagrangian Grassmannians, following recent work of Closset-Khlaif on Schubert cycles in ordinary Grassmannians. In the case of Lagrangian Grassmannians, there are superpotential terms in both the bulk GLSM as well as on the defect itself, enforcing isotropy constraints. We check our construction by comparing the locus on which the GLSM defect is supported to mathematical descriptions, checking dimensions, and perhaps most importantly, comparing defect indices to known and expected polynomial invariants of the Schubert cycles in quantum cohomology and quantum K theory. 

Nakada’s colored hook formula is a vast generalization of many important formulae in combinatorics, such as the classical hook length formula and the Peterson’s formula for the number of reduced expressions of minuscule Weyl group elements. In this paper, we use cohomological properties of Segre–MacPherson classes of Schubert cells and varieties to prove a generalization of a cohomological version of Nakada’s formula, in terms of smoothness properties of Schubert varieties. A key ingredient in the proof is the study of a decorated version of the Bruhat graph. Weights of the paths in this graph give the terms in the generalized Nakada’s formula, and the summation over all paths is equal to the equivariant multiplicity of the Chern–Schwartz–MacPherson class of a Richardson variety. Among the applications we mention an algorithm to calculate structure constants of multiplications of Segre–MacPherson classes of Schubert cells, and a skew version of Nakada–Peterson’s formula. 

Abstract. The Peterson variety is a subvariety of the flag manifold G/B equipped with an action of a one-dimensional torus, and a torus invariant paving by affine cells, called Peterson cells. We prove that the equivariant pull-backs of Schubert classes indexed by arbitrary Coxeter elements are dual (up to an intersection multiplicity) to the fundamental classes of Peterson cell closures. Dividing these classes by the intersec- tion multiplicities yields a Z-basis for the equivariant cohomology of the Peterson variety. We prove several properties of this basis, including a Graham positivity property for its structure constants, and stability with respect to inclusion in a larger Peterson variety. We also find for- mulae for intersection multiplicities with Peterson classes. This explains geometrically, in arbitrary Lie type, recent positivity statements proved in type A by Goldin and Gorbutt.