Notes on Schubert, Grothendieck and Key Polynomials

Abstract

Abstract. We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.Key words: plactic monoid and reduced plactic algebras; nilCoxeter and idCoxeter algebras; Schubert, β-Grothendieck, key and (double) key-Grothendieck, and Di FrancescoZinn-Justin polynomials; Cauchy's type kernels and symmetric, totally symmetric plane partitions, and alternating sign matrices; noncrossing Dyck paths and (rectangular) Schubert polynomials; multi-parameter deformations of Genocchi numbers of the first and the second types; Gandhi-Dumont polynomials and (staircase) Schubert polynomials; double affine nilCoxeter algebras

On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

“…Definition 4. 71. The even generic Orlik-Solomon algebra OS + (Γ n ) is defined to be an associative algebra (say over Z) generated by the set of mutually commuting elements y i,j , 1 ≤ i = j ≤ n, subject to the set of cyclic relations…”

“…• If A = (−1, 2, 0, 1, 1), then S A w (X n ) is equal to the Di Francesco-Zinn-Justin polynomials and studied in [32,33,34] and [71].…”

“…• If A = (0, 0, 1, 0, 0), then K A α (X n ) is equal to the reduced key polynomial introduced in [71].…”

“…71. For any permutation w ∈ S n there exists a graph Γ w = (V, E), possibly with multiple edges, such that the reduced volume vol(F Γw ) of the flow polytope F Γw , see, e.g., [132] for a definition of the former, is equal to S w (1).…”

Odd supersymmetrization of elliptic R-matrices

“…In particular, equation (1.8) is fulfilled by the properly normalized Baxter-Belavin elliptic R-matrix [13], which is then treated as a matrix generalization of the Kronecker function (1.3). Applications of (1.8) can be found in [6,10].…”