Quantum cohomology of Grassmannians and cyclotomic fields
Quantum cohomology of Grassmannians and cyclotomic fields
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DOI：
10.1070/rm2006v061n01abeh004304
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发表时间：
2006-02
影响因子：
0.9
通讯作者：
Sergey Galkin;Vasilii Viktorovich Golyshev
中科院分区：
文献类型：
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作者：
Sergey Galkin;Vasilii Viktorovich Golyshev
关于格拉斯曼的量子上同调，见[1]。给定整数l， N与1 6 l 0 < σλ, σμ, σν∨> d q σν，其中ν是ν的对偶。设ζ是（- 1）的原始n次方根。令K = Q(ζ), K = N−1，令Λ = Z[e1, e2，…]]是对称函数的（分级）环，其中ei是第i个初等对称函数。设它是第i个完全对称函数。令E(t) = P (t) H(t) = P (t) hit，我们得到E(- t)H(t) = 1。将对称函数σ应用于元组（x1，…）的结果。， xn)的参数将用σ（x1，…）表示。xn)。最后，我们输入Λ ' = Λ/（el+1, el+2，…）， ΛQ = Λ⊗Q， ΛQ = Λ '⊗Q和ΛK = ΛQ⊗Q K.定理1 （Siebert-Tian[2]）。由ST (q) = 1⊗q, ST (ei) = ci(S)⊗1定义的环ST： Λ[q]→QH（G，Z）的同态，且KerST = (el+1, el+2，…；hN−1 +1，…， hN−1,hN +(−1)q)。令QH(G，Q) = QH(G，Z)⊗Q为有理量子上同环，令QH（G，Q, 1）表示其在Q = 1时的专一性。令I1 = (hN−1 +1)，…， hN−1,hN +(−1))，
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)),