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Damir Yeliussizov - One of the best experts on this subject based on the ideXlab platform.
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Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs
Journal of Combinatorial Theory Series A, 2018Co-Authors: Damir YeliussizovAbstract:Abstract Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.
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Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs
arXiv: Combinatorics, 2017Co-Authors: Damir YeliussizovAbstract:Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. This approach allows us to prove skew Cauchy identity which is our central result. We then derive various consequences and applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also explain the finite expansion property for products of Grothendieck polynomials.
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Duality and deformations of stable Grothendieck polynomials
Journal of Algebraic Combinatorics, 2016Co-Authors: Damir YeliussizovAbstract: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.
Kim, Jang Soo - One of the best experts on this subject based on the ideXlab platform.
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Refined canonical stable Grothendieck polynomials and their duals
2021Co-Authors: Hwang Byung-hak, Jang Jihyeug, Kim, Jang Soo, Song Minho, Song U-keunAbstract:In this paper we introduce refined canonical stable Grothendieck polynomials and their duals with two infinite sequences of parameters. These polynomials unify several generalizations of Grothendieck polynomials including canonical stable Grothendieck polynomials due to Yeliussizov, refined Grothendieck polynomials due to Chan and Pflueger, and refined dual Grothendieck polynomials due to Galashin, Liu, and Grinberg. We give Jacobi--Trudi-type formulas, combinatorial models, Schur expansions, Schur positivity, and dualities of these polynomials. We also consider flagged versions of Grothendieck polynomials and their duals with skew shapes.Comment: 55 pages, 10 figure
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Jacobi--Trudi formula for refined dual stable Grothendieck polynomials
2021Co-Authors: Kim, Jang SooAbstract:In 2007 Lam and Pylyavskyy found a combinatorial formula for the dual stable Grothendieck polynomials, which are the dual basis of the stable Grothendieck polynomials with respect to the Hall inner product. In 2016 Galashin, Grinberg, and Liu introduced refined dual stable Grothendieck polynomials by putting additional sequence of parameters in the combinatorial formula of Lam and Pylyavskyy. Grinberg conjectured a Jacobi--Trudi type formula for refined dual stable Grothendieck polynomials. In this paper this conjecture is proved by using bijections of Lam and Pylyavskyy.Comment: 25 pages, 25 figure
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Jacobi--Trudi formulas for flagged refined dual stable Grothendieck polynomials
2020Co-Authors: Kim, Jang SooAbstract:Recently Galashin, Grinberg, and Liu introduced the refined dual stable Grothendieck polynomials, which are symmetric functions in $x=(x_1,x_2,\dots)$ with additional parameters $t=(t_1,t_2,\dots)$. The refined dual stable Grothendieck polynomials are defined as a generating function for reverse plane partitions of a given shape. They interpolate between Schur functions and dual stable Grothendieck polynomials introduced by Lam and Pylyavskyy in 2007. Flagged refined dual stable Grothendieck polynomials are a more refined version of refined dual stable Grothendieck polynomials, where lower and upper bounds are given for the entries of each row or column. In this paper Jacobi--Trudi-type formulas for flagged refined dual stable Grothendieck polynomials are proved using plethystic substitution. This resolves a conjecture of Grinberg and generalizes a result by Iwao and Amanov--Yeliussizov.Comment: 32 pages, 4 figure
Matilde Marcolli - One of the best experts on this subject based on the ideXlab platform.
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Graph hypersurfaces and a dichotomy in the Grothendieck ring
Letters in Mathematical Physics, 2011Co-Authors: Paolo Aluffi, Matilde MarcolliAbstract:The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of Z generated by the class of a point. This clarifies the extent to which the graph hypersurfaces ‘generate the Grothendieck ring of varieties’: while it is known that graph hypersurfaces generate the Grothendieck ring over a localization of Z[L] in which L becomes invertible, the span of the graph hypersurfaces in the Grothendieck ring itself is nearly killed by setting the Lefschetz motive L to zero. In particular, this shows that the graph hypersurfaces do not generate the Grothendieck ring prior to localization. The same result yields some information on the mixed Hodge structures of graph hypersurfaces, in the form of a constraint on the terms in their Deligne–Hodge polynomials. These observations are certainly not surprising for the expert reader, but are somewhat hidden in the literature. The treatment in this note is straightforward and self-contained.
Steimle Wolfgang - One of the best experts on this subject based on the ideXlab platform.
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Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings
2021Co-Authors: Calmès Baptiste, Dotto Emanuele, Harpaz Yonatan, Hebestreit Fabian, Land Markus, Moi Kristian, Nardin Denis, Nikolaus Thomas, Steimle WolfgangAbstract:We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring $R$ to the homotopy $\mathrm{C}_2$-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in $R$ from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of $\mathbb{Z}$, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated, and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of global dimension $d$ is an equivalence in degrees $\geq d+3$. As an important tool, we establish the hermitian analogue of Quillen's localisation-d\'evissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi.Comment: 56 pages This version: minor improvements, updated reference
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Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings
HAL CCSD, 2020Co-Authors: Calmès Baptiste, Dotto Emanuele, Harpaz Yonatan, Hebestreit Fabian, Land Markus, Moi Kristian, Nardin Denis, Nikolaus Thomas, Steimle WolfgangAbstract:56 pagesWe establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring $R$ to the homotopy $\mathrm{C}_2$-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in $R$ from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of $\mathbb{Z}$, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated, and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of global dimension $d$ is an equivalence in degrees $\geq d+3$. As an important tool, we establish the hermitian analogue of Quillen's localisation-d\'evissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi
Shinsuke Iwao - One of the best experts on this subject based on the ideXlab platform.
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Free-fermions and skew stable Grothendieck polynomials.
arXiv: Combinatorics, 2020Co-Authors: Shinsuke IwaoAbstract:Skew stable Grothendieck polynomials are $K$-theoretic analogues of skew Schur polynomials. We give a free-fermionic presentation of skew stable Grothendieck polynomials and their dual symmetric functions. By using our presentation, we derive a family of determinantal formulas, which are $K$-analogues of the Jacobi-Trudi formula for skew Schur functions. We also introduce a combinatorial method to calculate certain expansions of skew (dual) stable Grothendieck polynomials by using the non-commutative supersymmetric Schur functions.
