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Schubert polynomials for the affine Grassmannian. (English) Zbl 1149.05045
Let \(G\) be a simply-connected simple algebraic group over \(\mathbb{C}\) and let \(K\subset C\) be a maximal compact subgroup with maximal torus \(T\subset K\). Let \({\mathcal G}r_G\) be the affine Grassmannian associated to \(G\). Then, \({\mathcal G}r_G\) is homotopy equivalent to the based continuous loop group \(\Omega_e(K)\). Let \(W\) be the (finite) Weyl group of \(G\) and let \(W_{\text{aff}}\) be the corresponding affine Weyl group. For any \(w\in W_{\text{aff}}/W\), let \(\Omega_w\subset{\mathcal G}r_G\) be the Bruhat cell. We denote by \(\sigma_w\in H_*({\mathcal G}r_G,\mathbb{Z})\) (resp., \(\sigma^w\in H^*({\mathcal G}r_G,\mathbb{Z})\)) the corresponding Schubert homology (resp., cohomology) class. Since \(\Omega_e(K)\) is a topological group, \(H_*({\mathcal G}r_G,\mathbb{Z})\) and \(H^*({\mathcal G}r_G,\mathbb{Z})\) are dual Hopf algebras.
Let \(\mathbb{A}_{\text{aff}}\) denote the Kostant-Kumar nil-Hecke ring associated to \(W_{\text{aff}}\). There is a canonical embedding of the \(T\)-equivariant cohomology \(S:= H^T(pt)\) of a point into \(\mathbb{A}_{\text{aff}}\). Then, as shown by Peterson, the centralizer \(Z_{\mathbb{A}_{\text{aff}}}(S)\) of \(S\) in \(\mathbb{A}_{\text{aff}}\) is isomorphic with the \(T\)-equivariant homology \(H_T({\mathcal G}r_G,\mathbb{Z})\). The \(\mathbb{Z}\)-algebra \(\mathbb{A}_{\text{aff}}\otimes_S\mathbb{Z}\) admits a subalgebra \(\mathbb{B}'\) called the affine Fomin-Stanley subalgebra. Lam shows that \(\mathbb{B}'\) is isomorphic with \(Z_{\mathbb{A}_{\text{aff}}}(S)\otimes_S\mathbb{Z}\); in particular, by Peterson’s result, \(\mathbb{B}'\) is isomorphic with \(H_*({\mathcal G}r_G,\mathbb{Z})\) as a Hopf algebra. As a corollary of this, one obtains that \(\mathbb{B}'\) is a commutative algebra.
When \(G= SL_n(\mathbb{C})\), Bott identified \(H^*({\mathcal G}r_G,\mathbb{Z})\) (resp., \(H^*({\mathcal G}r_G,\mathbb{Z})\)) with a certain subring (resp., a certain quotient ring) of the ring \(\Lambda_{\mathbb{Z}}\) of symmetric functions over \(\mathbb{Z}\) in infinitely many variables. The principal result of this paper of Lam identifies the bases \(\sigma_w\) and \(\sigma^w\) as elements of \(\Lambda_{\mathbb{Z}}\). These are, respectively, the \(k\)-Schur functions (introduced by Lapointe-Lascoux-Morse) and the affine Schur functions (introduced by Lapointe-Morse and also Lam). In the homology this was conjectured by M. Shimozono and in the cohomology this was conjectured by J. Morse.
The above identifications allow one to conclude that the product structure constants for the \(k\)-Schur functions as well as the affine Schur functions are nonnegative integers by virtue of the corresponding nonnegativity results for the product of the homology Schubert classes \(\{\sigma_w\}\) (due to Peterson) and the product of the cohomology Schubert classes \(\{\sigma^w\}\) (due to Kumar-Nori).

MSC:
05E05 Symmetric functions and generalizations
14N15 Classical problems, Schubert calculus
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