Quantum cohomology of Grassmannians and cyclotomic fields

Abstract

For the quantum cohomology of Grassmannians, see [1]. Given integers l, N with 1 6 l 0〈σλ, σμ, σν∨〉d q σν , where ν ∨ is the partition dual to ν. Let ζ be a primitive Nth root of (−1). Put K = Q(ζ), k = N − l and let Λ = Z[e1, e2, . . . ] be the (graded) ring of symmetric functions, where ei is the ith elementary symmetric function. Let hi be the ith complete symmetric function. Putting E(t) = P eit , H(t) = P hit , we have E(−t)H(t) = 1. The result of applying a symmetric function σ to a tuple (x1, . . . , xn) of arguments will be denoted by σ(x1, . . . , xn). Finally, we put Λ ′ = Λ/(el+1, el+2, . . . ), ΛQ = Λ ⊗ Q, ΛQ = Λ ′ ⊗ Q and ΛK = ΛQ ⊗Q K. Theorem 1 (Siebert–Tian [2]). The homomorphism of rings ST: Λ[q]→QH(G,Z) defined by ST (q) = 1 ⊗ q, ST (ei) = ci(S) ⊗ 1 is an epimorphism, and KerST = (el+1, el+2, . . . ;hN−l+1, . . . , hN−1, hN + (−1)q). Let QH(G,Q) = QH(G,Z) ⊗ Q be the rational quantum cohomology ring and let QH(G,Q, 1) denote its specialization for q = 1. Putting I1 = (hN−l+1, . . . , hN−1, hN + (−1)),