On q-symmetric functions and q-quasisymmetric functions

Abstract

In this paper, we construct the q-analogue of Poirier-Reutenauer algebras, related deeply with other q-combinatorial Hopf algebras. As an application, we use them to realize the odd Schur functions defined by Ellis, Khovanov, and Lauda, then naturally obtain the odd Littlewood-Richardson rule concerned by Ellis. Moreover, we construct the refinement of the odd Schur functions, called odd quasisymmetric Schur functions, parallel to the consideration by Haglund, Luoto, Mason, and van Willigenburg. All the q-Hopf algebras we discuss here provide the corresponding q-dual graded graphs.

Notes on Schubert, Grothendieck and Key Polynomials

“…Now let q = 0, ±1 be a parameter, and assume that q ij = q, ∀ 1 ≤ i < j ≤ n − 1. This case has been treated recently in [55]. We expect that the generalized Knuth relations (5.3) are related with quantum version of the tropical/geometric RSK-correspondence (work in progress), and, as expected, with a q-weighted version of the Robinson-Schensted algorithm, presented in [60].…”

“…We also mention and leave for a separate publication(s), the case of algebras and polynomials associated with superplactic monoid [38,56], which corresponds to the relations SPL q with q i = 1, ∀ i. Finally we point out an interesting and important paper [55] wherein the case Z = ∅, and the all deformation parameters are equal to each other, has been independently introduced and studied in depth.…”

Simultaneous and sensitive detection of dopamine and uric acid using a poly(L-methionine)/gold nanoparticle-modified glassy carbon electrode

“…Gold nanoparticles (nano-Au) can bind strongly to the surface of some polymers through covalent bonding to functional groups such as CN, NH2, and SH [34][35][36]. Polymers including 3-methylthiophene, thiophene, and methionine possess S atoms in their structure, which can readily form strong chemical bonds with nano-Au [37,38].…”

The shifted Poirier–Reutenauer algebra

“…One simple counterexample is as follows. (ii) In [12] one of us has defined the q-analogue of PR algebras and applied that to study the odd Schur functions introduced by Ellis, Khovanov and Lauda. We expect there should also exist an odd counterpart for Schur's P-functions.…”