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关于典型群的双Schubert多项式和Grothendieck多项式

On Double Schubert and Grothendieck polynomials for Classical Groups

A. N. Kirillov

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摘要：

本文给出了与经典型$A,B, C$和$D$李群的满旗簇对应的双Schubert多项式和双$β$-Grothendieck多项式的（非交换）生成函数的代数组合构造.我们的方法是基于某些"转移矩阵“的分解对应的指数解的量子杨-巴克斯特方程的NiCoxeter或IdCoxeter代数的经典类型。我们引入的“三元”$β$多项式${\mathfrak{G}}_{w}^{W}(X,Y,Z)$满足相干条件和（广义）消失条件.它们的生成函数在代数$Id_βCoxeter(W)$中有一个很好的因式分解，因此，多项式${\mathfrak{G}}_{w}^{W}(X,Y,Z)$允许用$W$型白日梦的形式进行组合描述。

英文摘要：

We give an algebra-combinatorial constructions of (noncommutative) generating functions of double Schubert and double $β$-Grothendieck polynomials corresponding to the full flag varieties associated to the Lie groups of classical types $A,B, C$ and $D$. Our approach is based on the decomposition of certain `` transfer matrices `` corresponding to the exponential solution to the quantum Yang--Baxter equations associated with either NiCoxeter or IdCoxeter algebras of classical types. The "triple"~$β$-Grothendieck polynomials ${\mathfrak{G}}_{w}^{W}(X,Y,Z)$ we have introduced, satisfy, among other things, the coherency and (generalized) vanishing conditions. Their generating function has a nice factorization in the algebra $Id_βCoxeter(W)$, and as a consequence, the polynomials ${\mathfrak{G}}_{w}^{W}(X,Y,Z)$ admit a combinatorial description in terms of $W$-type pipe dreams.

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