arXiv subjects

Young-Hun Kim

Publications and source records attributed to Young-Hun Kim.

12 recordsLinked to original sources

arXiv2026

Weak Bruhat interval modules of the 0-Hecke algebras for stable Grothendieck polynomials

For a partition $\lambda$, let $G_\lambda^{(\beta)}$ be the stable $\beta$-Grothendieck polynomial attached to $\lambda$. Each homogeneous component of the $\beta = 1$ specialization $G_\lambda^{(1)}$ is Schur-positive and hence positive in the fundamental basis of quasisymmetric functions. For $m\ge|\lambda|$, let $G_{\lambda,m}^{(1)}$ be the homogeneous degree $m$ component of $G_\lambda^{(1)}$. In this paper, we first give a direct proof of an expansion of $G_{\lambda,m}^{(1)}$ in the fundamental basis in terms of standard set-valued tableaux. We then use these tableaux as a basis to define a module of the $0$-Hecke algebra and show that the quasisymmetric characteristic of the resulting module is $G_{\lambda,m}^{(1)}$. We further show that this module decomposes as a direct sum of weak Bruhat interval modules.

math.RT↗

arXiv2026

A representation-theoretic interpretation of the Schur expansion of two-row genomic Schur functions

Genomic Schur functions were introduced by Pechenik and Yong in connection with the $K$-theory of Grassmannians. Pechenik proved that genomic Schur functions admit a positive expansion in the basis of fundamental quasisymmetric functions and, for partitions with two parts, a positive expansion in the Schur basis. Later, Kim and Yoo constructed $0$-Hecke modules associated with genomic Schur functions and conjectured that the latter expansion admits a representation-theoretic interpretation in terms of $0$-Hecke modules. In this paper, we prove the conjecture of Kim and Yoo, thereby obtaining a representation-theoretic interpretation of the Schur expansion in the two-row case.

math.CO↗

arXiv2026

Poset modules of the $0$-Hecke algebras of type $B$

In 2001, Chow developed the theory of the $B_n$ posets $P$ and the type $B$ $P$-partition enumerators $K^B_P$. To provide a representation-theoretic interpretation of $K^B_P$, we define the poset modules $M^B_P$ of the 0-Hecke algebra $H_n^B(0)$ of type $B$ by endowing the set of type-$B$ linear extensions of $P$ with an $H_n^B(0)$-action. We then show that the Grothendieck group of the category associated to type-$B$ poset modules is isomorphic to the space of type $B$ quasisymmetric functions as both a $\mathrm{QSym}$-module and comodule, where $\mathrm{QSym}$ denotes the Hopf algebra of quasisymmetric functions. Considering an equivalence relation on $B_n$ posets, where two posets are equivalent if they share the same set of type-$B$ linear extensions, we identify a natural representative of each equivalence class, which we call a distinguished poset. We further characterize the distinguished posets whose sets of type-$B$ linear extensions form intervals in the right weak Bruhat order on the the hyperoctahedral groups. Finally, we discuss the relationship among the categories associated to type-$B$ weak Bruhat interval modules, $B_n$ poset modules, and finite-dimensional $H_n^B(0)$-modules.

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arXiv2023

Regular Schur labeled skew shape posets and their 0-Hecke modules

Assuming Stanley's $P$-partition conjecture holds, the regular Schur labeled skew shape posets with underlying set $\{1,2,\ldots, n\}$ are precisely the posets $P$ such that the $P$-partition generating function is symmetric and the set of linear extensions of $P$, denoted $\Sigma_L(P)$, is a left weak Bruhat interval in the symmetric group $\mathfrak{S}_n$. We describe the permutations in $\Sigma_L(P)$ in terms of reading words of standard Young tableaux when $P$ is a regular Schur labeled skew shape poset, and classify $\Sigma_L(P)$'s up to descent-preserving isomorphism as $P$ ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the $0$-Hecke modules $\mathsf{M}_P$ associated with regular Schur labeled skew shape posets $P$ up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the posets whose linear extensions form a dual plactic-closed subset of $\mathfrak{S}_n$. Using this characterization, we construct distinguished filtrations of $\mathsf{M}_P$ with respect to the Schur basis when $P$ is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the $0$-Hecke modules $\mathsf{M}_P$ are also discussed.

math.RT↗

arXiv2023

Poset modules of the $0$-Hecke algebras and related quasisymmetric power sum expansions

Duchamp--Hivert--Thibon introduced the construction of a right $H_n(0)$-module, denoted as $M_P$, for any partial order $P$ on the set $[n]$. This module is defined by specifying a suitable action of $H_n(0)$ 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 $\bigoplus_{n \ge 0} G_0(\mathscr{P}(n))$ has a Hopf algebra structure that is isomorphic to the Hopf algebra of quasisymmetric functions, where $\mathscr{P}(n)$ is the full subcategory of $\textbf{mod-}H_n(0)$ whose objects are direct sums of finitely many isomorphic copies of poset modules and $G_0(\mathscr{P}(n))$ is the Grothendieck group of $\mathscr{P}(n)$. 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 $Y_\alpha$ of Schur functions, where $\alpha$ 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 tableaux.

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arXiv2022

Weak Bruhat interval modules for genomic Schur functions

Let $\lambda$ be a partition of a positive integer $n$. The genomic Schur function $U_\lambda$ was introduced by Pechenik--Yong in the context of the $K$-theory of Grassmannians. Recently, Pechenik provided a positive combinatorial formula for the fundamental quasisymmetric expansion of $U_\lambda$ in terms of increasing gapless tableaux. In this paper, for each $1 \le m \le n$, we construct an $H_m(0)$-module $\mathbf{G}_{\lambda;m}$ whose image under the quasisymmetric characteristic is the $m$th degree homogeneous component of $U_\lambda$ by defining an $H_m(0)$-action on increasing gapless tableaux. We provide a method to assign a permutation to each increasing gapless tableau, and use this assignment to decompose $\mathbf{G}_{\lambda;m}$ into a direct sum of weak Bruhat interval modules. Furthermore, we determine the projective cover of each summand of the direct sum decomposition.

math.RT↗

arXiv2022

Homological properties of 0-Hecke modules for dual immaculate quasisymmetric functions

Let $n$ be a nonnegative integer. For each composition $\alpha$ of $n$, Berg $\textit{et al.}$ introduced a cyclic indecomposable $H_n(0)$-module $\mathcal{V}_\alpha$ with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study $\mathcal{V}_\alpha$'s from the homological viewpoint. To be precise, we construct a minimal projective presentation of $\mathcal{V}_\alpha$ and a minimal injective presentation of $\mathcal{V}_\alpha$ as well. Using them, we compute ${\rm Ext}^1_{H_n(0)}(\mathcal{V}_\alpha, {\bf F}_\beta)$ and ${\rm Ext}^1_{H_n(0)}( {\bf F}_\beta, \mathcal{V}_\alpha)$, where ${\bf F}_\beta$ is the simple $H_n(0)$-module attached to a composition $\beta$ of $n$. We also compute ${\rm Ext}_{H_n(0)}^i(\mathcal{V}_\alpha,\mathcal{V}_{\beta})$ when $i=0,1$ and $\beta \le_l \alpha$, where $\le_l$ represents the lexicographic order on compositions.

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arXiv2021

Weak Bruhat interval modules of the 0-Hecke algebra

The purpose of this paper is to provide a unified method for dealing with various 0-Hecke modules constructed using tableaux so far. To do this, we assign a $0$-Hecke module to each left weak Bruhat interval, called a weak Bruhat interval module. We prove that every indecomposable summand of the $0$-Hecke modules categorifying dual immaculate quasisymmetric functions, extended Schur functions, quasisymmetric Schur functions, and Young row-strict quasisymmetric Schur functions is a weak Bruhat interval module. We further study embedding into the regular representation, induction product, restriction, and (anti-)involution twists of weak Bruhat interval modules.

math.RT↗

arXiv2020

The projective cover of tableau-cyclic indecomposable $H_n(0)$-modules

Let $\alpha$ be a composition of $n$ and $\sigma$ a permutation in $\mathfrak{S}_{\ell(\alpha)}$. This paper concerns the projective covers of $H_n(0)$-modules $\mathcal{V}_\alpha$, $X_\alpha$ and $\mathbf{S}^\sigma_{\alpha}$, which categorify the dual immaculate quasisymmetric function, the extended Schur function, and the quasisymmetric Schur function when $\sigma$ is the identity, respectively. First, we show that the projective cover of $\mathcal{V}_\alpha$ is the projective indecomposable module $\mathbf{P}_\alpha$ due to Norton, and $X_\alpha$ and the $\phi$-twist of the canonical submodule $\mathbf{S}^{\sigma}_{\beta,C}$ of $\mathbf{S}^\sigma_{\beta}$ for $(\beta,\sigma)$'s satisfying suitable conditions appear as $H_n(0)$-homomorphic images of $\mathcal{V}_\alpha$. Second, we introduce a combinatorial model for the $\phi$-twist of $\mathbf{S}^\sigma_{\alpha}$ and derive a series of surjections starting from $\mathbf{P}_\alpha$ to the $\phi$-twist of $\mathbf{S}^{\mathrm{id}}_{\alpha,C}$. Finally, we construct the projective cover of every indecomposable direct summand $\mathbf{S}^\sigma_{\alpha, E}$ of $\mathbf{S}^\sigma_{\alpha}$. As a byproduct, we give a characterization of triples $(\sigma, \alpha, E)$ such that the projective cover of $\mathbf{S}^\sigma_{\alpha, E}$ is indecomposable.

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arXiv2020

Modules of the $0$-Hecke algebra arising from standard permuted composition tableaux

We study the $H_n(0)$-module $\mathbf{S}^\sigma_\alpha$ due to Tewari and van Willigenburg, which was constructed using new combinatorial objects called standard permuted composition tableaux and decomposed into cyclic submodules. First, we show that every direct summand appearing in their decomposition is indecomposable and characterize when $\mathbf{S}^\sigma_\alpha$ is indecomposable. Second, we find characteristic relations among $\mathbf{S}^\sigma_\alpha$'s and expand the image of $\mathbf{S}^\sigma_\alpha$ under the quasi characteristic in terms of quasisymmetric Schur functions. Finally, we show that the canonical submodule of $\mathbf{S}^\sigma_\alpha$ appears as a homomorphic image of a projective indecomposable module.

math.RT↗

arXiv2020

Embedding linear codes into self-orthogonal codes and their optimal minimum distances

We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary $k$-dimensional linear code $\mathcal{C}$ ($k = 2,3,4$) into a self-orthogonal code of the shortest length which has the same dimension $k$ and minimum distance $d' \ge d(\mathcal{C})$. For $k > 4$, we suggest a recursive method to embed a $k$-dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length $n$ with dimension 4 and any length $n \not\equiv 6,13,14,21,22,28,29 \pmod{31}$ with dimension 5. We determine the exact optimal minimum distances of $[n,4]$ self-orthogonal codes which were left open by Li-Xu-Zhao (2008) when $n \equiv 0,3,4,5,10,11,12 \pmod{15}$. Then, using MAGMA, we observe that our embedding sends an optimal linear code to an optimal self-orthogonal code.

cs.IT↗

arXiv2019

Cyclic sieving phenomenon on dominant maximal weights over affine Kac-Moody algebras

We construct a (bi)cyclic sieving phenomenon on the union of dominant maximal weights for level $\ell$ highest weight modules over an affine Kac-Moody algebra with exactly one highest weight being taken for each equivalence class, in a way not depending on types, ranks and levels. In order to do that, we introduce $\textbf{\textit{S}}$-evaluation on the set of dominant maximal weights for each highest modules, and generalize Sagan's action by considering the datum on each affine Kac-Moody algebra. As consequences, we obtain closed and recursive formulae for cardinality of the number of dominant maximal weights for every highest weight module and observe level-rank duality on the cardinalities.

math.RT↗