\(K\)-theoretic polynomials. (English) Zbl 1507.05102
Summary: We contribute to modern symmetric function theory, introducing two new bases of the polynomial ring and studying their relations to known bases. The quasiLascoux basis is a \(K\)-theoretic deformation of the quasikey basis that is also a nonsymmetric lift of quasiGrothendieck polynomials. We give positive expansions of this new basis into the glide and Lascoux atom bases, as well as of the Lascoux basis into this new basis. These results include the first proof that quasiGrothendieck polynomials expand positively in the multifundamental quasisymmetric polynomials of T. Lam and P. Pylyavskyy [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)]. The kaons are \(K\)-theoretic deformations of fundamental particles. We give positive expansions of the glide and Lascoux atom bases into this new kaon basis. Throughout, we explore parallels between these \(K\)-analogues and their cohomological counterparts.
MSC:
| 05E05 | Symmetric functions and generalizations |
| 05E10 | Combinatorial aspects of representation theory |
Citations:
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