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 $Id_{ \beta}Coxeter(W)$, and as a consequence, the polynomials ${ \mathfrak{ G}}_{ w}^{ W}(X,Y,Z)$ admit a combinatorial description in terms of $W$-type pipe dreams.
Categories: math.CO