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

[摘要] We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations.

[关键词] braid and Yang–Baxter groups;classical and dynamical Yang–Baxter relations;classical Yang–Baxter;Kohno–Drinfeld and 3-term relations algebras;Dunkl;Gaudin and Jucys–Murphy elements;small quantum cohomology and K-theory of flag varieties;Pieri rules;Schubert;Grothendieck;Schr¨oder;Ehrhart;Chromatic;Tutte and Betti polynomials;reduced polynomials;Chan–Robbins–Yuen polytope;k-dissections of a convex (n + k + 1)- gon;Lagrange inversion formula and Richardson permutations;multiparameter deformations of Fuss–Catalan and Schr¨oder polynomials;Motzkin;Riordan;Fine;poly-Bernoulli and Stirling numbers;Euler numbers and Brauer algebras;VSASM and CSTCPP;Birman– Ko–Lee monoid;Kronecker elliptic sigma functions