On Double Schubert and Grothendieck polynomials for Classical Groups

Abstract

We give an algebra-combinatorial constructions of (noncommutative) generating functions of double Schubert and double $\beta$-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"~$\beta$-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 $I{d}_{\beta }Coxeter(W)$, and as a consequence, the polynomials ${\mathfrak{G}}_w^W(X,Y,Z)$ admit a combinatorial description in terms of $W$-type pipe dreams.

29p

References in corpus (1)

• Degeneracy Loci, Pfaffians, and Vexillary Signed Permutations in Types B, C, and D

Cited by in corpus (2)

• Comodule Structures, Equivariant Hopf Structures, and Generalized Schubert Polynomials
• Cauchy Formulas and Billey's Formulas for Generalized Grothendieck polynomials