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

Clinical and Pathological Characteristics of Four Korean Patients with Limb-Girdle Muscular Dystrophy type 2B

Limb-girdle muscular dystrophy type 2B (LGMD2B), a subtype of autosomal recessive limb-girdle muscular dystrophy (ARLGMD), is characterized by a relatively late onset and slow progressive course. LGMD2B is known to be caused by the loss of the dysferlin protein at sarcolemma in muscle fibers. In this study, the clinical and pathological characteristics of Korean LGMD2B patients were investigated. Seventeen patients with ARLGMD underwent muscle biopsy and the histochemical examination was performed. For the immunocytochemistry, a set of antibodies against dystrophin, α, β, γ, δ-sarcoglycans, dysferlin, caveolin-3, and β-dystroglycan was used. Four patients (24%) showed selective loss of immunoreactivity against dysferlin at the sarcolemma on the muscle specimens. Therefore, they were classified into the LGMD2B category. The age at the onset of disease ranged from 9 yr to 33 yr, and none of the patients was wheelchair bound at the neurological examination. The serum creatine kinase (CK) was high in all the patients (4010-5310 IU/L). The pathologic examination showed mild to moderate dystrophic features. These are the first Korean LGMD2B cases with a dysferlin deficiency confirmed by immunocytochemistry. The clinical, pathological, and immunocytochemical findings of the patients with LGMD2B in this study were in accordance with those of other previous reports

Dissolution of Spinel Oxides and Capacity Losses in 4 V Li/LixMn2O4 Cells

Dissolution of spinel manganese oxides and the concomitant cathodic capacity losses were examined in 4 V Li/PC + DME + LiClO4/LiMn2O4 cells where PC is propylene carbonate and DME is dimethoxyethane. Dissolved Mn2 contents in the electrolytes were analyzed as a function of cathode potential and carbon contents in the composite cathodes. Characteristically, manganese dissolution was notably high at the charged state (at >4.1 V vs. Li/Lit), in which potential range an electrochemical oxidation of the solvent molecules was also prominent. From this and another observation whereby the Mn dissolution increased with increasing carbon content in the composite cathodes, it was proposed that, at the charged state of the cathode the solvent molecules are electrochemically oxidized on carbon surfaces and an as-generated species promotes the manganese dissolution. Results of an ac impedance study revealed that Mn dissolution brings about an increase in contact resistances at the Mn-depleted spinel/carbon interface, and also in the electrode reaction resistances for Li intercalation/deintercalation. Thus, the Mn dissolution causes capacity losses in two different pathways; material loss of the loaded spinel and polarization loss due to a cell resistance increment. The former prevailed when cathodes contained excess amounts of carbon, while the latter became more of a problem as the carbon contents decreased

Cost-effectiveness of a medication event monitoring system for tuberculosis management in Morocco

BACKGROUND: Digital health technologies have been used to enhance adherence to TB medication, but the cost-effectiveness remains unclear. METHODS: We used the real data from the study conducted from April 2014 to December 2020 in Morocco using a smart pillbox with a web-based medication monitoring system, called Medication Event Monitoring Systems (MEMS). Cost-effectiveness was evaluated using a decision analysis model including Markov model for Multi-drug resistant (MDR) TB from the health system perspective. The primary outcome was the incremental cost-effectiveness ratio (ICER) per disability adjusted life-year (DALY) averted. Two-way sensitive analysis was done for the treatment success rate between MEMS and standard of care. RESULTS: The average total per-patient health system costs for treating a new TB patient under MEMS versus standard of care were $398.70 and$155.70, respectively. The MEMS strategy would reduce the number of drug-susceptible TB cases by 0.17 and MDR-TB cases by 0.01 per patient over five years. The ICER of MEMS was $434/DALY averted relative to standard of care, and was most susceptible to the TB treatment success rate of both strategies followed by the managing cost of MEMS. CONCLUSION: MEMS is considered cost-effective for managing infectious active TB in Morocco