Anatol N. Kirillov, “On Some Quadratic Algebras I $\frac{1}{2}$: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss–Catalan, Universal Tutte and Reduced Polynomials”, SIGMA, 12 (2016), 002, 172 pp.

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Symmetry, Integrability and Geometry: Methods and Applications, 2016, Volume 12, 002, 172 pp.
DOI: https://doi.org/10.3842/SIGMA.2016.002 (Mi sigma1084)

This article is cited in 12 scientific papers (total in 12 papers)

On Some Quadratic Algebras I

\frac{1}{2}

$\frac{1}{2}$ : 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

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DOI: https://doi.org/10.3842/SIGMA.2016.002

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

Keywords: braid and Yang–Baxter groups; classical and dynamical Yang–Baxter relations; classical Yang–Baxter, Kohno–Drinfeld and

3

$3$ -term relations algebras; Dunkl, Gaudin and Jucys–Murphy elements; small quantum cohomology and

K

$K$ -theory of flag varieties; Pieri rules; Schubert, Grothendieck, Schröder, Ehrhart, Chromatic, Tutte and Betti polynomials; reduced polynomials; Chan–Robbins–Yuen polytope;

k

$k$ -dissections of a convex

(n + k + 1)

$(n+k+1)$ -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.

Received: March 23, 2015; in final form December 27, 2015; Published online January 5, 2016

Bibliographic databases:

Document Type: Article

MSC: 14N15; 53D45; 16W30

Language: English

Citation: Anatol N. Kirillov, “On Some Quadratic Algebras I

\frac{1}{2}

$\frac{1}{2}$ : Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss–Catalan, Universal Tutte and Reduced Polynomials”, SIGMA, 12 (2016), 002, 172 pp.

Citation in format AMSBIB

\Bibitem{Kir16}

\by Anatol~N.~Kirillov

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

\jour SIGMA

\yr 2016

\vol 12

\papernumber 002

\totalpages 172

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Linking options:

https://www.mathnet.ru/eng/sigma1084

https://www.mathnet.ru/eng/sigma/v12/p2

This publication is cited in the following 12 articles:

Shinsuke Iwao, “Free-fermions and skew stable Grothendieck polynomials”, J Algebr Comb, 56:2 (2022), 493
A. Levin, M. Olshanetsky, A. Zotov, “Odd supersymmetric Kronecker elliptic function and Yang-Baxter equations”, J. Math. Phys., 61:10 (2020), 103504
Buciumas V., Scrimshaw T., Weber K., “Colored Five-Vertex Models and Lascoux Polynomials and Atoms”, J. Lond. Math. Soc.-Second Ser., 102:3 (2020), 1047–1066
A. Levin, M. Olshanetsky, A. Zotov, “Odd supersymmetrization of elliptic r-matrices”, J. Phys. A-Math. Theor., 53:18 (2020), 185202
Shinsuke Iwao, “Grothendieck polynomials and the boson-fermion correspondence”, Algebraic Combinatorics, 3:5 (2020), 1023
I. A. Sechin, A. V. Zotov, “ ${\rm GL}_{NM}$ quantum dynamical $R$ -matrix based on solution of the associative Yang–Baxter equation”, Russian Math. Surveys, 74:4 (2019), 767–769
A. Grekov, I. Sechin, A. Zotov, “Generalized model of interacting integrable tops”, J. High Energy Phys., 2019, no. 10, 081
T. Hudson, T. Matsumura, “Vexillary degeneracy loci classes in $K$ -theory and algebraic cobordism”, Eur. J. Comb., 70 (2018), 190–201
Darij Grinberg, “ $t$ -Unique Reductions for Mészáros's Subdivision Algebra”, SIGMA, 14 (2018), 078, 34 pp.
Seung Jin Lee, “Chern class of Schubert cells in the flag manifold and related algebras”, J Algebr Comb, 47:2 (2018), 213
T. Matsumura, “An algebraic proof of determinant formulas of Grothendieck polynomials”, Proc. Jpn. Acad. Ser. A-Math. Sci., 93:8 (2017), 82–85
Damir Yeliussizov, “Duality and deformations of stable Grothendieck polynomials”, J Algebr Comb, 45:1 (2017), 295

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Symmetry, Integrability and Geometry: Methods and Applications

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