Poset modules of the 00-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.Comment: 42 page

The projective cover of tableau-cyclic indecomposable Hn(0)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.Comment: 41 page

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.Comment: 44 pages, to be published in Forum of Math: Sigm

Semi-automatic Filtering of Translation Errors in Triangle Corpus

The meaning that Justice has after a conflict in a society might vary regarding the political development and cultural and shared values of a certain society. Rawls, in his Theory of Justice gives his idea of what justice is and presents two principles of justice that he argues are required to live in a good society: a first principle that secures equal rights and liberties for all individuals and a second egalitarian principle that restrains the consequences of economic inequalities within societies. He also introduces the concept of “overlapping consensus” which I will use regarding the idea of Reconciliation, at the end of this paper. In the cases presented in this paper (i.e. Argentina and South Africa), essential human rights were violated, therefore wrongdoers made the society unjust. The aim of this thesis is to elucidate the conditions that are necessary to re-establish justice when a society goes through a conflict. I will introduce some ideas concerning that issue: ideas of retribution, reparation and reconciliation. These are seen as different paths for several countries when trying to tackle to the matter of achieving justice. In my view, this question can be answered appealing first to an intuitive conception of moral justice that may exist at an individual and collective level, as well. The ethical dilemmas both levels have are in relation to the harm done, punishments and how to balance them, limiting, for instance, the punishment in order to accomplish a just and a better society. I will also present how shared values can result from a process of reconciliation, which is considered as the ideal alternative to achieve justice. However, when the equilibrium between members of a community is broken, some people claim that punishment can restore that lost equilibrium that existed before in the community. Nevertheless, peace, reconciliation and justice cannot be constructed under the basis of silence. One way to keep memory alive is to let survivors, for instance, narrate what they have lived through; telling stories also creates a new space to share with others their experiences, revealing their fears and emotions. Regarding this theme, I will present the NUNCA MAS (Never Again) report, which is fundamental as it gives some testimonies, facts and proposals that will help to reach a consensus and therefore, future reconciliations. Why is important to achieve justice? Because then members of a certain community will be able to interact in the present with common shared values and thus, deal with the past. Not to consider reconciliation as one important step to achieve justice, and only think in terms of punishment, instead of giving way to peace and justice, could perhaps promote the possibility of further conflicts. To consider both punishment and reconciliation might just be one possible blueprint in the long and difficult way of searching for a just society

PREFACE

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