On equivariant quantum Schubert calculus for G/P

Abstract

We show a Z 2 -filtered algebraic structure and a "quantum to classical" principle on the torus-equivariant quantum cohomology of a complete flag variety of general Lie type, generalizing earlier works of Leung and the second author. We also provide various applications on equivariant quantum Schubert calculus, including an equivariant quantum Pieri rule for partial flag variety F ℓ n 1 ,··· ,n k ;n+1 of Lie type A.

W-translated Schubert divisors and transversal intersections

“…There is another type of identities among (equivariant) genus zero, three-point Gromov-Witten invariants for complete flag variety G/B of general Lie types in [29,16]. The special case of degree zero Gromov-Witten invariants recovers the recurrence formula in [22].…”

Equivariant Pieri rules for isotropic Grassmannians

“…Namely, we prove that any Pieri coefficient N µ λ,p (Gr(m, N )) is equal to a Pieri coefficient of the form N ν ν,p ′ (Gr(m ′ , N )) in a possibly different Grassmannian Gr(m ′ , N ). Such a reduction has also been made in [21] and [42] in a combinatorial way, but our argument is much simpler and can explain its geometric origin. Each coefficient N ν ν,p ′ (Gr(m ′ , N )) is the restriction of a special equivariant Schubert class [X p ′ ] T to a T -fixed point of Gr(m ′ , N ).…”

An equivariant quantum Pieri rule for the Grassmannian on cylindric shapes

“…More precisely, given a pair of partitions µ Ď λ P P mn such that the skew shape λ{µ " v r , Huang and Li define a new diagram λ µ in the shorter rectangle P m´r,n . The closed formula for λ µ is provided in [HL,Definition 3.14], and interpreted through a join-and-cut process explained in [HL,Definition 3.15]. We now review the construction of λ µ , using a slight reformulation of the join-and-cut algorithm from [HL].…”

“…In comparison to other known equivariant quantum Pieri and/or Littlewood-Richardson rules, Theorem 1.1 more directly captures the quantum-to-affine phenomenon which governs the ring structure of QH T pGrpm, nqq via the parabolic Peterson isomorphism, as explicated in [CM] in the non-equivariant case. In addition, each of the existing positive combinatorial formulas for these products typically requires doing calculations in many different related two-step flags or smaller Grassmannians in order to calculate a single Pieri product; compare [HL,BM,Buc2]. We emphasize, by contrast, that Theorem 1.1 permits an entire equivariant quantum Pieri product to be carried out directly on the skew shapes λ{d{µ as in Figure 1.1, which fully illustrates the product σ ‹ σ P QH T pGrp3, 5qq; see Example 2.14.…”