Equivariant Schubert calculus of Coxeter groups
The cohomology ring of the flag variety \(G/B\) is isomorphic to the coinvariant ring of the associated Weyl group \(W\). The equivariant cohomology ring of \(G/B\) with respect to the action of the maximal torus \(T\subset B\) is isomorphic to the double coinvariant ring of \(W\). On the other hand, GKM theory [\textit{M. Goresky} et al., Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)] gives another description of the equivariant cohomology ring \(H_T(G/B)\) as the image of the localization map \(\bigoplus_{w\in W}\iota^*_w: H_T(G/B)\rightarrow\bigoplus_{w\in W} H_T(wB/B)\cong\bigoplus_{w\in W}\text{Sym}(\mathfrak{t}^*)\). Based on this description, one can show that for each \(w\in W\), and each Schubert subvariety \(X_w\subset G/B\), there is a unique homogeneous equivariant cohomology class \(Z_w\in H_T(G/B)\) of degree \(2\ell(w)\) characterized by the following properties: (1) \(\iota_v^*\left(Z_w\right)=0\) for all \(\ell(v)\leq \ell(w)\) with \(v\neq w\), and (2) \(\iota_w^*\left(Z_w\right)=\prod_{\beta\in\Pi^+}\beta\), where \(\Pi^+\subset\mathfrak{t}^*\) is the set of positive roots in the root system corresponding to \(G\), \(B\), and \(T\). The main result in the paper under review is the definition of the double Schubert polynomials \(\mathfrak{S}_w(t,x)\) (\(w\in W\)) in the double coinvariant ring of an arbitrary finite Coxeter group, which, in the Weyl group case, represent the equivariant classes \(Z_w\) (\(w\in W\)) in the equivariant cohomology ring \(H_T(G/B)\). The author also proves an equivariant version of Chevalley's rule for his double Schubert polynomials. Along the way the author shows that the double coinvariant ring of a finite Coxeter group \(W\) is isomorphic to the GKM ring of \(W\)--an algebraic analogue of GKM description of \(H_T(G/B)\) above. While this result itself is certainly not new, it is hard to find a proof explicitly written down anywhere in the literature (coincidentally, another proof of this result was given independently by this reviewer in his PhD. thesis). As a corollary, the author obtains the polynomial values of his double Schubert polynomials \(\mathfrak{S}_w(t,x)\) at all ``fixed points'' \(v\in W\), recovering a well known result of \textit{S. C. Billey} [Duke Math. J. 96, No. 1, 205--224 (1999; Zbl 0980.22018)].
scientific article; zbMATH DE number 6342481
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