On Double Schubert and Grothendieck polynomials for Classical Groups

2015/04/07 by A. N. Kirillov, Kirillov, A. N. · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1504.01469

29p

arxiv created 2015/04/07 · arxiv updated 2015/04/08

Abstract

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 \mathfrakG_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 \mathfrakG_w^W(X,Y,Z) admit a combinatorial description in terms of W-type pipe dreams.

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• Classical double Grothendieck transitions

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