For a finite dimensional algebra $A$ over field $k$, the 2-term silting complexes of gives simplicial complex $\Delta(A)$ called \emph{$g$-simplicial complex}. We give tilting theoretic interpretations $h$-vectors and Dehn-Sommerville equations $\Delta(A)$. Using $g$-vectors complexes, nonsingular fan $\Sigma(A)$ in real Grothendieck group $K_0({\rm proj} A)_\mathbb{R}$ \emph{$g$-fan}. several basic properties including sign-coherence, sign decomposition, idempotent reductions, Jasso...

In this paper, we introduce a new generating function called $d$-polynomial for the dimensions of $\tau$-tilting modules over given finite dimensional algebra. Firstly, study basic properties $d$-polynomials and show that it can be realized as certain sum $f$-polynomials simplicial complexes arising from $\tau$-rigid pairs. Secondly, give explicit formulas preprojective algebras path Dynkin quivers by using close relation with $W$-Eulerian polynomials $W$-Narayana polynomials. Thirdly,...

For Brauer graph algebras, tilting mutation is compatible with flip of graphs. The aim this paper to generalize result the class configuration algebras introduced by Green and Schroll recently. More precisely, under a certain condition, we introduce configurations prove that it corresponding algebras.

In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across filtration. this work, we extend setting, propose use bipath homology, which can be used to study persistence pair filtrations connected at their ends, compare two filtrations. interval-decomposability is guaranteed, provide an algorithm for computing diagrams discuss interpretation diagrams.

Abstract In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across filtration. this work, we extend setting, propose use bipath , which can be used to study persistence pair filtrations connected at their ends, compare two filtrations. interval-decomposability is guaranteed, provide an algorithm for computing diagrams discuss interpretation diagrams.

To study the set of torsion classes a finite dimensional basic algebra, we use decomposition, called sign-decomposition, parametrized by elements $\{\pm1\}^n$ where $n$ is number simple modules. If $A$ an algebra with radical square zero, then for each $\epsilon \in \{\pm1\}^n$ there hereditary $A_{\epsilon}^!$ zero and bijection between associated to $\epsilon$ faithful $A_{\epsilon}^!$. Furthermore, this preserves property being functorially finite. As application in $\tau$-tilting theory,...

In this paper, we determine the $τ$-tilting finiteness for some blocks of (classical) Schur algebras. Combining with results in arXiv:2010.05206, get a complete classification finite As refinement, also give algebra $S(2,r)$.

The $g$-fan of a finite dimensional algebra is fan in its real Grothendieck group defined by tilting theory. We give classification complete $g$-fans rank 2. More explicitly, our first main result asserts that every sign-coherent 2 some algebra. Our proof based on three fundamental results, Gluing Theorem, Rotation Theorem and Subdivision which realize basic operations fans the level algebras. For each 16 convex $\Sigma$ 2, second gives characterization algebras $A$ satisfying...

Recently, there is growing interest in the use of relative homology algebra to develop invariants using interval covers and resolutions (i.e., right minimal approximations interval-decomposable modules) for multi-parameter persistence modules. In this paper, set all modules over a given poset plays central role. Firstly, we show that restriction each indecomposable direct summand injective. This result suggests way simplify computation covers. Secondly, monotonicity resolution global...

In this paper, we compute the number of two-term tilting complexes for an arbitrary symmetric algebra with radical cube zero over algebraically closed field. Firstly, give a complete list algebras having only finitely many isomorphism classes in terms their associated graphs. Secondly, enumerate each case list.

The $g$-vectors of two-term presilting complexes are important invariants. We study a fan consisting all $g$-vector cones for complete gentle algebra. show that any algebra is $g$-tame, by definition, the closure geometric realization its entire ambient vector space. Our main ingredients their surface model and asymptotic behavior under Dehn twists. On other hand, it known special biserial factor $g$-tameness preserved taking algebras. As consequence, we get

We show that any Brauer tree algebra has precisely $\binom{2n}{n}$ $2$-tilting complexes, where $n$ is the number of edges associated tree. More explicitly, for an external edge $e$ and integer $j\neq0$, we complexes $T$ with $g_e(T)=j$ $\binom{2n-|j|-1}{n-1}$, $g_e(T)$ denotes $e$-th $g$-vector $T$. To prove this, use a geometric model graph algebras on closed oriented marked surfaces classification due to Adachi-Aihara-Chan.