Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 12 (2016), 002, 172 pages      arXiv:1502.00426      https://doi.org/10.3842/SIGMA.2016.002

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

Anatol N. Kirillov abc
a) Research Institute of Mathematical Sciences (RIMS), Kyoto, Sakyo-ku 606-8502, Japan
b) The Kavli Institute for the Physics and Mathematics of the Universe (IPMU), 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
c) Department of Mathematics, National Research University Higher School of Economics, 7 Vavilova Str., 117312, Moscow, Russia

Received March 23, 2015, in final form December 27, 2015; Published online January 05, 2016

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

Key words: 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, Schröder, 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 Schröder 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.

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