19 problems

Chapoton's conjecture. The incidence algebras $k[J(\Phi^+)]$ are fractionally Calabi--Yau.

Let $(X,\leqslant)$ be a locally finite pre-ordered set, let $\mathcal{R}$ be a $2$-torsion-free commutative ring with unity, and let $I(X,\mathcal{R})$ denote the incidence algebr…

Frattini conjecture. The global dimension of $IA(\operatorname{Sub}_G)$ is the largest integer $n$ for which there exists a subgroup $H$ satisfying

Incidence-algebra discrepancy conjecture. There exists a group $G$ for which the global dimension of the rational incidence algebra for ${\operatorname{Sub}_G}/G$ differs from the…

Let $P$ be a finite poset and $KP$ its incidence algebra. An incidence algebra is Coxeter-independent when its Coxeter permutation does not depend on the natural labelling. DJMSSTW…

Let $L$ be a lattice and let $A=kL$ be its incidence algebra. The paper introduces a combinatorial version of periodic Serre formality for such algebras. Combinatorial–periodic Ser…

Let $P$ be a finite poset on a finite set. A poset is called conclusive if it is either soluble, meaning that every transitive system on $P$ can be completed for every Abelian grou…

Let $L$ be a finite lattice, and let $A=KL$ be its incidence algebra. Assume that $A$ has a pure minimal injective coresolution. Distributivity conjecture. Then $L$ is distributive…

Let $K$ be a field of characteristic $0$, and let $P$ be a poset. Denote its incidence algebra by $KP$, and let independent and the Coxeter permutation be the corresponding notions…

Tameness conjecture for g-tame incidence algebras. If $\operatorname{Inc}_\K(P)$ is $\mathbf{g}$-tame, then $\operatorname{Inc}_\K(P)$ is tame.

Wildness and g-tameness conjecture. The incidence algebra $\operatorname{Inc}_\K(P)$ is wild if and only if it is not $\mathbf{g}$-tame.

Let $X$ be a finite connected poset, let $I(X,K)$ be its incidence algebra over $K$, and let $D(X,K)$ denote the diagonal subalgebra. A map $\theta:I(X,K)\to I(X,K)$ is a strong co…

Let $(P,\leq)$ be a finite poset. Denote by $Z_n$ the partially ordered set whose Hasse diagram is a zig-zag orientation of an affine quiver of type $\mathbb{A}_n$. Lattice conject…

Let $P$ be a finite poset, and let $\field P$ denote its incidence algebra. For the critical and regular levels $a_0,t_1,a_1,t_2,\dots,t_n,a_n$ associated with the Morse function…

Let $X$ be a finite poset, let $\mathbb{F}$ be a field of characteristic zero, and let $G$ be a group. Consider a $G$-grading on the incidence algebra $I(X)$. For each minimal idem…

Conjectured convolution-inverse formula. For the relevant indices $a,b,c,d$,

Let $K\Delta$ be a PHI algebra derived equivalent to the category of coherent sheaves on a weighted projective line, and let $M$ be its canonical sincere $K\Delta$-module. Here, a…

Let $K\Delta$ be a piecewise hereditary incidence algebra (PHI algebra), and let $HH^1(K\Delta)$ denote its first Hochschild cohomology group. An algebra is simply connected when i…

Universality conjecture. $U$ is universal for complete decomposition spaces and cULF maps, in the sense that complete decomposition spaces are classified by cULF maps into $U$.