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Combinatorial \(B_ n\)-analogues of Schubert polynomials. (English) Zbl 0871.05060

The paper under review deals with type \(B\) analogues of the Schubert polynomials of Lascoux and Schützenberger. Let \(W=W_n\) be the Coxeter group of type \(A_n\) (i.e. the symmetric group \(S_{n+1}\)) with the natural action of \(W\) on \({\mathbb C}[x_1,\ldots,x_{n+1}]\). Let \(I_W\) be the ideal generated by the \(W\)-symmetric polynomials without constant terms. The Schubert polynomials \(X_w\) are laballed with the elements \(w\in W\) and are a distingushed linear basis of \({\mathbb C}[x_1,\ldots,x_{n+1}]/I_W\).
The authors choose five properties in the “ordinary” \(A\) case: (0) \(X_w\) is homogeneous of degree the length \(l(w)\) of \(w\) with respect to the natural set of generators \(s_i\) of \(W\). (1) \(X_w\) can be recursively defined by divided difference operators by \(\partial_i(X_{ws_i})=X_w\) if \(l(ws_i)=l(w)+1\). (2) For any \(u,v\in W\) and for a sufficiently large \(m\), in the multiplication of \(X_u\) and \(X_v\) one can get rid of the unpleasant “mod \(I_W\)”, i.e. \(X_uX_v=\sum_{w\in W_m}c_{uv}^wX_w\). (3) The polynomials \(X_w\) are with nonnegative integer coefficients. (4) The Schubert polynomials are stable with respect to the natural embedding \(W_n\subset W_m\), \(n<m\), and the corresponding projection \[ {\mathbb C}[x_1,\ldots,x_{m+1}]\longrightarrow {\mathbb C}[x_1,\ldots,x_{n+1}] \] sending the extra variables to 0.
The authors show that it is impossible to transfer all these properties to the case of the hyperoctahedral group \(B_n\), i.e. the group of symmetries of the \(n\)-dimensional cube. They require the properties (0) and (4) and construct \(B_n\)-Schubert polynomials satisfying also the conditions (1)-(2) and (2)-(3). The constructions are based on the exponential solution of the Yang-Baxer equation in the nil-Coxeter algebra . The authors show that the two kinds of Schubert polynomials are related by a certain “change of variables”. Finally, they construct a family of polynomials \(X_w\) and conjecture that they satisfy the conditions (0), (1), (3) and (4). Recently the conjecture was solved affirmatively by Tao Kai Lam. Other interesting results involving, e.g., Stanley symmetric functions of type \(B\), are also obtained in this paper.
Reviewer: V.Drensky (Sofia)

MSC:

05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E05 Symmetric functions and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
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