Excited Young diagrams and equivariant Schubert calculus

Abstract

Abstract. We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call "excited Young diagrams" and the second one is written in terms of factorial Schur Q-or P -functions. As an application, we give a Giambelli-type formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety.

Hilbert functions of points on Schubert varieties in orthogonal Grassmannians

“…The papers quoted in the previous paragraph and [3] give some answers in the special cases they consider. More interestingly, the recent article [6] gives not only an interpretation of the multiplicity similar to ours but also a closed form formula. But it is not clear that the approach of [6] leads to Gröbner degenerations or for that matter to the Hilbert function: the point of [6] is to circumvent the need for Gröb-ner degenerations (precursors of [6] in the Grassmannian and symplectic Grassmannian cases having had these as starting points), while ours is to arrive at them.…”

“…The points of OR(v) are those that are (strictly) above the diagonal, and the points of N(v) are those that are to the South-West of the poly-line captioned 'boundary of N(v)': we draw the boundary so that points on the boundary belong to N(v). The reader can readily verify that d = 13 and v = (1,2,3,4,6,7,10,11,13,15,18,19,22) We will be considering monomials, also called multisets, in some of these sets. A monomial, as usual, is a subset with each member being allowed a multiplicity (taking values in the non-negative integers).…”

“…This finishes the description of the map Oφ. For β in S w,j,j +1 (up), we define the orthogonal piece-star O(T, w, β) corresponding to β as 3,4,5,6,9,10,13,14,17,18), w= (8,10,13,14,16,18,19,20,21,22,23,24) and T = {(21, 3), (16,4), (16,5), (7, 6)}. Figure 9 shows the elements of S w and T.…”

“…We have: 10,13,14,16,18,19,20,21,22,23,24), (20,5), (20,9), (16,9), (7,6), (8,6), (19,17), (19, 18)} Figure 11 shows these monomials. Clearly w is diagonal at 1, 3, 5; in other words δ 1 , δ 3 , δ 5 exist: δ 1 = (23, 1), δ 3 = (21, 3), and δ 5 = (16, 5).…”

“…But it is not clear that the approach of [6] leads to Gröbner degenerations or for that matter to the Hilbert function: the point of [6] is to circumvent the need for Gröb-ner degenerations (precursors of [6] in the Grassmannian and symplectic Grassmannian cases having had these as starting points), while ours is to arrive at them. In any case, [6] is different in both method and the nature of results relied upon.…”