arXiv subjects

Yu Qiu

Publications and source records attributed to Yu Qiu.

78 records · Page 5Linked to original sources

arXiv2015

Decomposition of Graphs into $(k,r)$-Fans and Single Edges

Let $ϕ(n,H)$ be the largest integer such that, for all graphs $G$ on $n$ vertices, the edge set $E(G)$ can be partitioned into at most $ϕ(n, H)$ parts, of which every part either is a single edge or forms a graph isomorphic to $H$. Pikhurko and Sousa conjectured that $ϕ(n,H)=\ex(n,H)$ for $χ(H)\geqs3$ and all sufficiently large $n$, where $\ex(n,H)$ denotes the maximum number of edges of graphs on $n$ vertices that does not contain $H$ as a subgraph. A $(k,r)$-fan is a graph on $(r-1)k+1$ vertices consisting of $k$ cliques of order $r$ which intersect in exactly one common vertex. In this paper, we verify Pikhurko and Sousa's conjecture for $(k,r)$-fans. The result also generalizes a result of Liu and Sousa.

math.CO↗

arXiv2015

Exchange graphs and Ext quivers

We study the oriented exchange graph $\textrm{EG}^\circ(Γ_{N}\,Q)$ of reachable hearts in the finite-dimensional derived category $\mathcal{D}(Γ_{N}\,Q)$ of the CY-$N$ Ginzburg algebra $Γ_{N}Q$ associated to an acyclic quiver $Q$. We show that any such heart is induced from some heart in the bounded derived category $\mathcal{D}(Q)$ via some `Lagrangian immersion' $\mathcal{L}:\mathcal{D}(Q)\to\mathcal{D}(Γ_{N}\,Q)$. We build on this to show that the quotient of $\textrm{EG}^\circ(Γ_{N}\,Q)$ by the Seidel-Thomas braid group is the exchange graph $\textrm{CEG}_{N-1}(Q)$ of cluster tilting sets in the (higher) cluster category $\mathcal{C}_{N-1}(Q)$. As an application, we interpret Buan-Thomas' coloured quiver for a cluster tilting set in terms of the Ext quiver of any corresponding heart in $\mathcal{D}(Γ_{N}\,Q)$.

math.RT↗

arXiv2015

C-sortable words as green mutation sequences

Let $Q$ be an acyclic quiver and $\mathbf{s}$ be a sequence with elements in the vertex set $Q_0$. We describe an induced sequence of simple (backward) tilting in the bounded derived category $\mathcal{D}(Q)$, starting from the standard heart $\mathcal{H}_Q=\operatorname{mod}\mathbf{k}Q$ and ending at another heart $\mathcal{H}_\mathbf{s}$ in $\mathcal{D}(Q)$. Then we show that $\mathbf{s}$ is a green mutation sequence if and only if every heart in this simple tilting sequence is greater than or equal to $\mathcal{H}_Q[-1]$; it is maximal if and only if $\mathcal{H}_\mathbf{s}=\mathcal{H}_Q[-1]$. This provides a categorical way to understand green mutations. Further, fix a Coxeter element $c$ in the Coxeter group $W_Q$ of $Q$, which is admissible with respect to the orientation of $Q$. We prove that the sequence $\widetilde{\mathbf{w}}$ induced by a $c$-sortable word $\mathbf{w}$ is a green mutation sequence. As a consequence, we obtain a bijection between $c$-sortable words and finite torsion classes in $\mathcal{H}_Q$. As byproducts, the interpretations of inversions, descents and cover reflections of a $c$-sortable word $\mathbf{w}$ are given in terms of the combinatorics of green mutations.

math.CO↗

arXiv2014

Tagged mapping class groups: Auslander-Reiten translation

We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface $\mathbf{S}$ with marked points and non-empty boundary, which generalizes Brüstle-Zhang's result for the puncture free case. As an application, we show that the intersection of the shifts in the 3-Calabi-Yau derived category $\mathcal{D}(Γ_{\mathbf{S}})$ associated to the surface and the corresponding Seidel-Thomas braid group of $\mathcal{D}(Γ_{\mathbf{S}})$ is empty, unless $\mathbf{S}$ is a polygon with at most one puncture (i.e. of type A or D).

math.RT↗

arXiv2014

Stability conditions and quantum dilogarithm identities for Dynkin quivers

We study fundamental group of the exchange graphs for the bounded derived category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category D(Γ_N Q) of the Calabi-Yau-N Ginzburg algebra associated to Q. In the case of D(Q), we prove that its space of stability conditions (in the sense of Bridgeland) is simply connected; as applications, we show that its Donanldson-Thomas invariant can be calculated via a quantum dilogarithm function on exchange graphs. In the case of D(Γ_N Q), we show that faithfulness of the Seidel-Thomas braid group action (which is known for Q of type A or N = 2) implies the simply connectedness of its space of stability conditions.

math.AG↗

arXiv2012

Ext-quivers of hearts of A-type and the orientation of associahedron

We classify the Ext-quivers of hearts in the bounded derived category D(A_n) and the finite-dimensional derived category D(Γ_N A_n) of the Calabi-Yau-N Ginzburg algebra D(Γ_N A_n). This provides the classification for Buan-Thomas' colored quiver for higher clusters of A-type. We also give explicit combinatorial constructions from a binary tree with n+2 leaves to a torsion pair in mod k\overrightarrow{A_n} and a cluster tilting set in the corresponding cluster category, for the straight oriented A-type quiver \overrightarrow{A_n}. As an application, we show that the orientation of the n-dimensional ssociahedron induced by poset structure of binary trees coincides with the orientation induced by poset structure of torsion pairs in mod k\overrightarrow{A_n} (under the correspondence above).

math.RT↗