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

Management of DiffServ-over-MPLS Transit Networks with BFD/OAM in ForCES Architecture †

Abstract. This paper proposes a management of DiffServ-over-MPLS transit network with BFD(Bidirectional Forwarding Detection)/OAM (operation, administration and maintenance) in ForCES (Forwarding and Control Element Separation) architecture for QoS-guaranteed DiffServ-over-MPLS traffic engineering. The proposed BFD and ForCES functions are implemented on Intel 2400 network processor, where BFD/OAM packets for MPLS TE-LSP are exchanged every 5 ~ 10 ms interval for performance measurements and link failure detection. The operations of BFD/OAM-based fault detection and performance measurement are controlled via distributed control plane with ForCES (forwarding and control element separation) architecture for large scale IP/MPLS router using multiple network processors in each network interface card. We explain the implementation details of ForCES-based distributed control plane functions, hierarchical traffic grooming with label stacking, and BFD/OAM mechanisms. The link failure detection performance of BFD/OAM functions for MPLS TE-LSP is evaluated

Observation of First-Order Metal-Insulator Transition without Structural Phase Transition in VO_2

An abrupt first-order metal-insulator transition (MIT) without structural phase transition is first observed by current-voltage measurements and micro-Raman scattering experiments, when a DC electric field is applied to a Mott insulator VO_2 based two-terminal device. An abrupt current jump is measured at a critical electric field. The Raman-shift frequency and the bandwidth of the most predominant Raman-active A_g mode, excited by the electric field, do not change through the abrupt MIT, while, they, excited by temperature, pronouncedly soften and damp (structural MIT), respectively. This structural MIT is found to occur secondarily.Comment: 4 pages, 4 figure

Association between volume status assessed by bioelectrical impedance analysis, lung ultrasound, or weight change and mortality in patients with sepsis-associated acute kidney injury receiving continuous kidney replacement therapy

Background Fluid overload is an independent risk factor of mortality in patients with acute kidney injury (AKI) receiving continuous kidney replacement therapy (CKRT). However, the association between fluid status, as assessed by bioelectrical impedance analysis (BIA) or lung ultrasound, and survival in patients with AKI requiring CKRT has not been established. Methods We analyzed 36 participants with sepsis-associated AKI who received CKRT at a tertiary hospital. The main exposures were volume surrogates: 1) overhydration normalized by extracellular water (OH/ECW, L/L) assessed by BIA, 2) the number of B-lines measured by lung ultrasound, and 3) weight change ([body weight at CKRT initiation – body weight at admission] × 100/body weight at admission). The primary outcome was the 28-day mortality. Results Seventeen participants (47.2%) died within 28 days. There were no significant correlations between OH/ECW and weight change (R2 = 0.040, p = 0.24), number of B-lines and OH/ECW (R2 = 0.056, p = 0.16), or weight change and number of B-lines (R2 = 0.014, p = 0.49). Kaplan-Meier analyses revealed that patients in the highest tertile of OH/ECW showed a significantly lower cumulative 28-day survival probability than the others (the lowest + middle tertiles). The survival probability of participants in the highest tertile of the number of B-lines or weight change did not differ from that of their counterparts. In a multivariate Cox proportional hazard model, the hazard ratio for the highest tertile of OH/ECW was 3.83 (95% confidence interval, 1.04–14.03). Conclusion Volume overload assessed using BIA (OH/ECW) was associated with the 28-day survival rate in patients with sepsis-associated AKI who received CKRT