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

DreamStyler: Paint by Style Inversion with Text-to-Image Diffusion Models

Recent progresses in large-scale text-to-image models have yielded remarkable accomplishments, finding various applications in art domain. However, expressing unique characteristics of an artwork (e.g. brushwork, colortone, or composition) with text prompts alone may encounter limitations due to the inherent constraints of verbal description. To this end, we introduce DreamStyler, a novel framework designed for artistic image synthesis, proficient in both text-to-image synthesis and style transfer. DreamStyler optimizes a multi-stage textual embedding with a context-aware text prompt, resulting in prominent image quality. In addition, with content and style guidance, DreamStyler exhibits flexibility to accommodate a range of style references. Experimental results demonstrate its superior performance across multiple scenarios, suggesting its promising potential in artistic product creation

Qualitative Anatomical Characteristics of Compression, Lateral, and Opposite Woods in Pinus merkusii and Agathis loranthifolia

This study aimed to observe andcompare the qualitative anatomical characteristics of compression (CW), lateral (LW), and opposite (OW) woodsin the stem wood of Sumatran pine (Pinus merkusii) and Agathis (Agathis loranthifolia). The anatomical characteristics were observed using optical microscopy and scanning electron microscopy. CW showed a gradual transition from earlywood to latewood in both species, circular tracheid shape, many intercellular spaces, irregular tracheid tips, helical cavities, and slit-like bordered pits. CW of Sumatran pine showed an indistinct growth ring, whileCW of Agathis showed a distinct growth ring. Helical ribs occurred only in CW of Sumatran pine. LW and OW showed an oval tracheid with an angular outline, regular tracheid arrangement, and tapered tracheid tips in both species. LW and OW showed mainly uniseriate bordered pits in Sumatran pine, while LW and OW of Agathis frequently showed multiseriatebordered pits. CW, LW, and OW showed fusiform and uniseriate rays in Sumatran pine, while those of Agathis showed uniseriate rays. In conclusion, CW showed distinctive qualitative anatomical characteristics to LW and OW in both species, while LW and OW mainly showed similar characteristics. In particular, there were considerably distinctive characteristics between CW from both species.Keywords: Agathis, anatomical characteristics, reaction wood, Sumatran pin