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Quasisymmetric Schur Q-functions and peak Young quasisymmetric Schur functions

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arXiv 2024年

作者： Choi, Seung-Il Nam, Sun-Young Oh, Young-Tak Center for Quantum Structures in Modules and Spaces Seoul National University Seoul08826 Korea Republic of Department of Mathematics Sogang University Seoul04107 Korea Republic of

In this paper, we explore the relationship between quasisymmetric Schur Q-functions and peak Young quasisymmetric Schur functions. We introduce a bijection on SPIT(α) such that {wc(T) | T ∈ SPIT(α)} and {wr(T) | T ∈ SPIT(α)} share identical descent distributions. Here, SPIT(α) is the set of standard peak immaculate tableaux of shape α, and wc and wr denote column reading and row reading, respectively. By combining this equidistribution with the algorithm developed by Allen, Hallam, and Mason, we demonstrate that the transition matrix from the basis of quasisymmetric Schur Q-functions to the basis of peak Young quasisymmetric Schur functions is upper triangular, with entries being non-negative integers. Furthermore, we provide explicit descriptions of the expansion of peak Young quasisymmetric Schur functions in specific cases, in terms of quasisymmetric Schur Q-functions. We also investigate the combinatorial properties of standard peak immaculate tableaux, standard Young composition tableaux, and standard peak Young composition tableaux. We provide a hook length formula for SPIT(α) and show that standard Young composition tableaux and standard peak Young composition tableaux can be bijectively mapped to specific words in a familiar form. Especially, cases of compositions with rectangular shape are examined in *** Codes 20C08, 05E05, 05E10 Copyright © 2024, The Authors. All rights reserved.

关键词： Combinatorial mathematics

POSET modules OF THE 0-HECKE ALGEBRAS AND RELATED QUASISYMMETRIC POWER SUM EXPANSIONS

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arXiv 2023年

作者： Choi, Seung-Il Kim, Young-Hun Oh, Young-Tak Center for Quantum structures in Modules and Spaces Seoul National University Seoul08826 Korea Republic of Department of Mathematics Sogang University Seoul04107 Korea Republic of

Duchamp–Hivert–Thibon introduced the construction of a right Hn(0)module, denoted as MP, for any partial order P on the set [n]. This module is defined by specifying a suitable action of Hn(0) on the set of linear extensions of P. In this paper, we refer to this module as the poset module associated with P. Firstly, we show that ⊕n≥0 G0(P(n)) has a Hopf algebra structure that is isomorphic to the Hopf algebra of quasisymmetric functions, where P(n) is the full subcategory of mod-Hn(0) whose objects are direct sums of finitely many isomorphic copies of poset modules and G0(P(n)) is the Grothendieck group of P(n). We also demonstrate how (anti-)automorphism twists interact with these modules, the induction product and restrictions. Secondly, we investigate the (type 1) quasisymmetric power sum expansion of some quasi-analogues Yα of Schur functions, where α is a composition. We show that they can be expressed as the sum of the P-partition generating functions of specific posets, which allows us to utilize the result established by Liu–Weselcouch. Additionally, we provide a new algorithm for obtaining these posets. Using these findings, for the dual immaculate function and the extended Schur function, we express the coefficients appearing in the quasisymmetric power sum expansions in terms of border strip *** Codes 20C08, 05E05, 05E10 Copyright © 2023, The Authors. All rights reserved.

关键词： Algebra

A COMPARISON OF CATEGORICAL AND TOPOLOGICAL ENTROPIES ON WEINSTEIN MANIFOLDS

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arXiv 2022年

作者： Bae, Hanwool Lee, Sangjin Center for Quantum Structures In Modules and Spaces Seoul National University Seoul Korea Republic of Pohang37673 Korea Republic of

Let W be a symplectic manifold, and let φ : W → W be a symplectic automorphism. This automorphism induces an auto-equivalence Φ defined on the Fukaya category of W . In this paper, we prove that the categorical entropy of Φ provides a lower bound for the topological entropy of φ, where W is a Weinstein manifold and φ is compactly supported. Furthermore, motivated by [cCGG24], we propose a conjecture that generalizes the result of [New88, Prz80, Yom87].MSC Codes 53D37, 53D40, 37J11 Copyright © 2022, The Authors. All rights reserved.

关键词： Topology

An exactly solvable model for emergence and scaling laws in the multitask sparse parity problem 24

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An exactly solvable model for emergence and scaling laws in ...

Proceedings of the 38th International Conference on Neural Information Processing Systems

作者： Yoonsoo Nam Nayara Fonseca Seok Hyeong Lee Chris Mingard Ard A. Louis Rudolf Peierls Centre for Theoretical Physics University of Oxford Center for Quantum Structures in Modules and Spaces Seoul National University Rudolf Peierls Centre for Theoretical Physics University of Oxford and Physical and Theoretical Chemistry Laboratory University of Oxford

ISBN: (纸本)9798331314385

关键词：

Lifting problem for universal quadratic forms over totally real cubic number fields

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arXiv 2023年

作者： Kim, Daejun Lee, Seok Hyeong School of Mathematics Korea Institute for Advanced Study Seoul02455 Korea Republic of Center for Quantum Structures in Modules and Spaces Seoul National University Seoul08826 Korea Republic of

Lifting problem for universal quadratic forms asks for totally real number fields K that admit a positive definite quadratic form with coefficients in Z that is universal over the ring of integers of K. In this paper, we show that K (Formula Presented) is the only such totally real cubic field. Moreover, we show that there is no such biquadratic *** Codes 11E12, 11R16, 11H06, 11H55 Copyright © 2023, The Authors. All rights reserved.

关键词： Number theory

CALABI–YAU structures ON RABINOWITZ FUKAYA CATEGORIES

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arXiv 2023年

作者： Bae, Hanwool Jeong, Wonbo Kim, Jongmyeong Center for Quantum Structures in Modules and Spaces Seoul National University Seoul08826 Korea Republic of Research Institute of Mathematics Seoul National University Seoul08826 Korea Republic of

In this paper, we prove that the derived Rabinowitz Fukaya category of a Liouville domain M of dimension 2n is (n − 1)-Calabi–Yau assuming the wrapped Fukaya category of M admits an at most countable set of Lagrangians that generate it and satisfy some finiteness condition on morphism spaces between *** Codes 53D37, 53D40, 18G70 Copyright © 2023, The Authors. All rights reserved.

关键词： Lagrange multipliers

An exactly solvable model for emergence and scaling laws in the multitask sparse parity problem

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arXiv 2024年

Deep learning models can exhibit what appears to be a sudden ability to solve a new problem as training time, training data, or model size increases, a phenomenon known as emergence. In this paper, we present a framework where each new ability (a skill) is represented as a basis function. We solve a simple multi-linear model in this skill-basis, finding analytic expressions for the emergence of new skills, as well as for scaling laws of the loss with training time, data size, model size, and optimal compute. We compare our detailed calculations to direct simulations of a two-layer neural network trained on multitask sparse parity, where the tasks in the dataset are distributed according to a power-law. Our simple model captures, using a single fit parameter, the sigmoidal emergence of multiple new skills as training time, data size or model size increases in the neural network. Copyright © 2024, The Authors. All rights reserved.

关键词： Contrastive Learning

HOMOLOGICAL PROPERTIES OF 0-HECKE modules FOR DUAL IMMACULATE QUASISYMMETRIC FUNCTIONS

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arXiv 2022年

作者： Choi, Seung-Il Kim, Young-Hun Nam, Sun-Young Oh, Young-Tak Center for Quantum Structures in Modules and Spaces Seoul National University Korea Republic of Department of Mathematics Sogang University Korea Republic of

Let n be a nonnegative integer. For each composition α of n, Berg, Bergeron, Saliola, Serrano, and Zabrocki introduced a cyclic indecomposable Hn(0)-module Vα with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study Vα’s from the homological viewpoint. To be precise, we construct a minimal projective presentation of Vα and a minimal injective presentation of Vα as well. Using them, we compute Ext1Hn(0)(Vα, Fβ) and Ext1Hn(0)(Fβ, Vα), where Fβ is the simple Hn(0)-module attached to a composition β of n. We also compute ExtiHn(0)(Vα, Vβ) when i = 0, 1 and β ≤l α, where ≤l represents the lexicographic order on *** Codes 20C08, 05E05, 05E10 Copyright © 2022, The Authors. All rights reserved.

关键词： Algebra

PSEUDO-ANOSOV AUTOEQUIVALANCES ARISING FROM SYMPLECTIC TOPOLOGY AND THEIR HYPERBOLIC ACTIONS ON STABILITY CONDITIONS

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arXiv 2023年

作者： Bae, Hanwool Lee, Sangjin Center for Quantum Structures in Modules and Spaces Seoul National University Seoul Korea Republic of Korea Institute for Advanced Study 85 Hoegiro Dongdaemun-gu Seoul02455 Korea Republic of

Within N-Calabi-Yau categories associated with quivers whose base graphs form trees, we delve into the study of the asymptotic behaviors of autoequivalences of a specific type. These autoequivalences, which we call "Penner type," exhibit straightforward asymptotic characteristics, making them noteworthy exemplars of "pseudo-Anosov" autoequivalences in the sense of [FFH+21], and also in a stronger sense that we define in the present paper. In addition, we provide a practical methodology for calculating the stretching factors of Penner type autoequivalences. We expect that this computational approach can have applications. As an example, we establish a positive lower bound on the translation length of the induced action these autoequivalences have on the space of stability conditions. Our anticipation is that this lower bound is, in fact, exact. Notably, we have observed instances of Penner type Φ where the induced actions align precisely with this lower bound. In other words, these examples induce hyperbolic actions on the space of stability conditions. Copyright © 2023, The Authors. All rights reserved.

关键词： Trees (mathematics)