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

Kirillov, Alexander

doi:10.3842/sigma.2016.002

Cited by 13 publications

(20 citation statements)

References 135 publications

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“…Proof. For G β λ , the formula (61) was given in [11] and hence implies here via β → α + β and x i → x i 1−αx i . The second formula is a dual version.…”

Section: Jacobi-trudi Type Identitiesmentioning

confidence: 99%

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Duality and deformations of stable Grothendieck polynomials

Yeliussizov

2016

J Algebr Comb

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Stable Grothendieck polynomials can be viewed as a K-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials, and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi-Trudi type identities, and associated Fomin-Greene operators.Comment: Journal of Algebraic Combinatorics, 201

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“…Proof. For G β λ , the formula (61) was given in [11] and hence implies here via β → α + β and x i → x i 1−αx i . The second formula is a dual version.…”

Section: Jacobi-trudi Type Identitiesmentioning

confidence: 99%

“…As a symmetric function, G λ has many similarities with s λ . For example, it can be defined by the following 'bi-alternant' formula [9,11,18] G λ (x 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Duality and deformations of stable Grothendieck polynomials

Yeliussizov

2016

J Algebr Comb

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“…We end with an overview of algebras similar to X /J that have appeared in the literature, making no claims of completeness. See also the last few paragraphs of the Introduction of [14] for a history of these algebras.…”

Section: A Deformation Of the Orlik-terao Algebra?mentioning

confidence: 99%

“…Why would one play a game like this? The reduction rule m → x i,k (x i,j +x j,k +β) x i,j x j,k m (this is a particular case of our above rule, when α is set to 0) has appeared in Karola Mészáros's study [17] of the abelianization of Anatol Kirillov's quasi-classical Yang-Baxter algebra (see, e.g., [14] for a recent survey of the latter and its many variants); it has a long prehistory (some of which is surveyed in Section 5.3 below), starting with Vladimir Arnold's 1971 work [2] on the braid arrangement. To define this abelianization 1 , we let β be an indeterminate (unlike in Example 1.1, where it was an element of Q).…”

Section: Introductionmentioning

confidence: 99%

“…seems to have appeared in work of Postnikov, Stanley and Mészáros; some concrete formulas (for specific values of the initial polynomial and specific values of β) appear in [26, Exercise A22] (resulting in Catalan and Narayana numbers). Recent work on Grothendieck polynomials by Anatol Kirillov (see [13,Section 4] and [14]) and by Laura Escobar and Karola Mészáros [7,Section 5] has again brought up the notion of substituting t i for x i,j in the polynomial obtained at the end of the game. This has led Mészáros to the conjecture that, after this substitution, the resulting polynomial no longer depends on the choices made during the game.…”

Section: Introductionmentioning

confidence: 99%

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t-Unique Reductions for Mészáros's Subdivision Algebra

Grinberg¹

2018

SIGMA

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Abstract. Fix a commutative ring k, two elements β ∈ k and α ∈ k and a positive integer n. Let X be the polynomial ring over k in the n(n − 1)/2 indeterminates x i,j for all 1 ≤ i < j ≤ n. Consider the ideal J of X generated by all polynomials of the form x i,j x j,k − x i,k (x i,j +x j,k +β)−α for 1 ≤ i < j < k ≤ n. The quotient algebra X /J (at least for a certain choice of k, β and α) has been introduced by Karola Mészáros in [Trans. Amer. Math. Soc. 363 (2011), 4359-4382] as a commutative analogue of Anatol Kirillov's quasi-classical YangBaxter algebra. A monomial in X is said to be pathless if it has no divisors of the form x i,j x j,k with 1 ≤ i < j < k ≤ n. The residue classes of these pathless monomials span the kmodule X /J , but (in general) are k-linearly dependent. More combinatorially: reducing a given p ∈ X modulo the ideal J by applying replacements of the form x i,j x j,k → x i,k (x i,j + x j,k + β) + α always eventually leads to a k-linear combination of pathless monomials, but the result may depend on the choices made in the process. More recently, the study of Grothendieck polynomials has led Laura Escobar and Karola Mészáros [Algebraic Combin. 1 (2018), 395-414] to defining a k-algebra homomorphism D from X into the polynomial ring k[t 1 , t 2 , . . . , t n−1 ] that sends each x i,j to t i . We show the following fact (generalizing a conjecture of Mészáros): If p ∈ X , and if q ∈ X is a k-linear combination of pathless monomials satisfying p ≡ q mod J , then D(q) does not depend on q (as long as β, α and p are fixed). Thus, the above way of reducing a p ∈ X modulo J may lead to different results, but all of them become identical once D is applied. We also find an actual basis of the k-module X /J , using what we call forkless monomials.

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Generalized model of interacting integrable tops

Grekov

Сечин

Zotov

2019

J. High Energ. Phys.

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We introduce a family of classical integrable systems describing dynamics of M interacting gl N integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the GL N R-matrix satisfying the associative Yang-Baxter equation. The obtained systems can be considered as extensions of the spin type Calogero-Moser models with (the classical analogues of) anisotropic spin exchange operators given in terms of the R-matrix data. In N = 1 case the spin Calogero-Moser model is reproduced. Explicit expressions for gl N M -valued Lax pair with spectral parameter and its classical dynamical r-matrix are obtained. Possible applications are briefly discussed.

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

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

Cited by 13 publications

References 135 publications

Duality and deformations of stable Grothendieck polynomials

Duality and deformations of stable Grothendieck polynomials

t-Unique Reductions for Mészáros's Subdivision Algebra

Generalized model of interacting integrable tops

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