A Demazure crystal construction for Schubert polynomials.

*(English)*Zbl 1390.14162Schubert polynomials \(\mathfrak{S}_w\) were first introduced by I. N. Bernstein et al. [Russ. Math. Surv. 28, No. 3, 1–26 (1973; Zbl 0289.57024)] as certain polynomial representatives of cohomology classes of Schubert cycles \(X_w\) in flag varieties. They were extensively studied by A. Lascoux and M.-P. Schützenberger [C. R. Acad. Sci., Paris, Sér. I 294, 447–450 (1982; Zbl 0495.14031)] using an explicit definition in terms of difference operators \({\partial}_w\). Subsequently, a combinatorial expression for Schubert polynomials as the generating polynomial for compatible sequences for reduced expressions of a permutation \(w\) was discovered by S. C. Billey et al. [J. Algebr. Comb. 2, No. 4, 345–374 (1993; Zbl 0790.05093)]. In the special case of the Grassmannian subvariety, Schubert polynomials are Schur polynomials, which also arise as the irreducible characters for the general linear group.

The Stanley symmetric functions \(F_w\) were introduced by R. P. Stanley [Eur. J. Comb. 5, 359–372 (1984; Zbl 0587.20002)] in the pursuit of enumerations of the reduced expressions of permutations, in particular of the long permutation \(w_0\). They are defined combinatorially as the generating functions of reduced factorizations of permutations. Stanley symmetric functions are the stable limit of Schubert polynomials \[ F_w (x_1, x_2, \ldots ) = \lim_{m\rightarrow \infty} \mathfrak{S}_{1^m \times w}(x_1, x_2, \ldots, x_{n+m}). \] P. Edelman and C. Greene [Adv. Math. 63, 42–99 (1987; Zbl 0616.05005)] showed that the coefficients of the Schur expansion of Stanley symmetric functions are nonnegative integer coefficients. Demazure modules for the general linear group are closely related to Schubert classes for the cohomology of the flag manifold. In certain cases these modules are irreducible polynomial representations, and so the Demazure characters also contain the Schur polynomials as a special case. Lascoux and Schützenberger stated that Schubert polynomials are nonnegative sums of Demazure characters.

In this paper the authors prove the converse to limit identity above by showing that Schubert polynomials are Demazure truncations of Stanley symmetric functions. Specifically, they show that the combinatorial objects underlying the Schubert polynomials, namely the compatible sequences, exhibit a Demazure crystal truncation of the full Stanley crystal of Morse and Schilling. They prove this, in which they give an explicit Demazure crystal structure on semi-standard key tableaux, which coincide with semi-skyline augmented fillings. Also they show that the crystal operators on reduced factorizations intertwine with (weak) Edelman-Greene insertion, proves their main result.

The Stanley symmetric functions \(F_w\) were introduced by R. P. Stanley [Eur. J. Comb. 5, 359–372 (1984; Zbl 0587.20002)] in the pursuit of enumerations of the reduced expressions of permutations, in particular of the long permutation \(w_0\). They are defined combinatorially as the generating functions of reduced factorizations of permutations. Stanley symmetric functions are the stable limit of Schubert polynomials \[ F_w (x_1, x_2, \ldots ) = \lim_{m\rightarrow \infty} \mathfrak{S}_{1^m \times w}(x_1, x_2, \ldots, x_{n+m}). \] P. Edelman and C. Greene [Adv. Math. 63, 42–99 (1987; Zbl 0616.05005)] showed that the coefficients of the Schur expansion of Stanley symmetric functions are nonnegative integer coefficients. Demazure modules for the general linear group are closely related to Schubert classes for the cohomology of the flag manifold. In certain cases these modules are irreducible polynomial representations, and so the Demazure characters also contain the Schur polynomials as a special case. Lascoux and Schützenberger stated that Schubert polynomials are nonnegative sums of Demazure characters.

In this paper the authors prove the converse to limit identity above by showing that Schubert polynomials are Demazure truncations of Stanley symmetric functions. Specifically, they show that the combinatorial objects underlying the Schubert polynomials, namely the compatible sequences, exhibit a Demazure crystal truncation of the full Stanley crystal of Morse and Schilling. They prove this, in which they give an explicit Demazure crystal structure on semi-standard key tableaux, which coincide with semi-skyline augmented fillings. Also they show that the crystal operators on reduced factorizations intertwine with (weak) Edelman-Greene insertion, proves their main result.

Reviewer: Cenap Özel (Bolu)

##### MSC:

14N15 | Classical problems, Schubert calculus |

05E10 | Combinatorial aspects of representation theory |

05A05 | Permutations, words, matrices |

05E05 | Symmetric functions and generalizations |

05E18 | Group actions on combinatorial structures |

20G42 | Quantum groups (quantized function algebras) and their representations |

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\textit{S. Assaf} and \textit{A. Schilling}, Algebr. Comb. 1, No. 2, 225--247 (2018; Zbl 1390.14162)

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