Shifted K-theoretic Poirier-reutenauer Algebra

Poirier and Reutenauer defined a Hopf algebra on the Z-span of all standard Young tableaux in [10], which is later studied in [4, 11]. The Robinson-Schensted-Knuth insertion was used to relate the bialgebra to Schur functions. Schur function is a class of symmetric functions that can be determined by the summation of all semistandard Young tableaux of shape . With the help of the PR-bialgebra, the Littlewood-Richardson rule is established, which gives an explicit description on the multiplication of arbitrary Schur functions. The generalization of this approach has been used to develop the Littlewood-Richardson rule for other classes of symmetric functions. In [9], a K-theoretic analogue is developed using Hecke insertion, providing a rule for multiplication of the stable Grothendieck polynomials. Similarly, in [6], a shifted analogue is developed, providing a rule for multiplication of P-Schur functions. We use a shifted Hecke insertion, introduced in [8], to develop a shifted K-theoretic version of the Poirier-Reutenauer algebra and an accompanying Littlewood-Richardson rule. Section 2 deals with the weak K-Knuth equivalence and its relationship with the shifted Hecke insertion. It is simultaneously a shifted analogue of Hecke insertion [1] and a Ktheoretic analogue of Sagan-Worley insertion in [12]. In section 3, we introduced a shifted K theoretic analogue of the Poirier-Reutenauer algebra which was first introduced in [10]. In section 4, we define the weak shifted stable Grothendieck polynomials. This is class of symmetric functions we are working with. Finally, section 5 introduces a LittlewoodRichardson rule of the weak shifted stable Grothendieck polynomials. The LittlewoodRichardson rule gives us an explicit description of the product structure of the weak shifted stable Grothendieck polynomials.