Flagged Grothendieck polynomials

Abstract

We show that the flagged Grothendieck polynomials defined as generating functions of flagged set-valued tableaux of Knutson et al. (J Reine Angew Math 630:1-31, 2009) can be expressed by a Jacobi-Trudi-type determinant formula generalizing the work of Hudson-Matsumura (Eur J Comb 70:190-201 2018). We also introduce the flagged skew Grothendieck polynomials in these two expressions and show that they coincide.

A Tableau Formula of Double Grothendieck Polynomials for 321-Avoiding Permutations

“…Recently, Anderson-Chen-Tarasca [2] extended the method in [10] to the case of 321-avoiding permutations in the study of Brill-Noether loci in K-theory, and obtained a determinant formula of the corresponding double Grothendieck polynomials. Combined with the work [15] by the author on skew Grothendieck polynomials, it implies a tableaux formula of (single) Grothendieck polynomials associated to 321 avoiding permutations. See also the most recent related work [5] by Chan-Pflueger.…”

“…If f 1 = · · · = f r , then the associated set-valued tableaux are nothing but the set-valued tableaux of skew shape λ/µ considered by Buch in [4]. Note that in [17] and [15], one considers more general flagged skew partitions and their set-valued tableaux.…”

Set-valued Rothe Tableaux and Grothendieck Polynomials

“…Such an idea was first used by Wachs [30] to establish the tableau formula for the Schubert polynomials of 2143-avoiding permutations. Matsumura [25] and Matsumura and Sugimoto [26] extended this idea to reprove the Knutson-Miller-Yong set-valued tableau formula for the Grothendieck polynomials of 2143-avoiding permutations. Our technique can be viewed as a generalization of that in [25,26] from Young diagrams to Rothe diagrams.…”

Refined canonical stable Grothendieck polynomials and their duals

“…Matsumura [31] introduced flagged Grothendieck polynomials as generating functions for flagged set-valued tableaux. These polynomials unify the Grothendieck polynomials and the flagged Schur functions, studied by Wachs [40].…”

“…(For explicit definitions, see Section 2.2.) Various Jacobi-Trudi formulas for generalizations of Grothendieck polynomials have been studied in the literature; see, e.g., [2,21,22,23,30,31,34,37,42].…”