Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFaithful actions of braid groups by twists along ADE-configurations of spherical objects
59
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove that the action of a generalized braid group on an enhanced triangulated categories, generated by spherical twist functors along an ADE-configuration of $omega$-spherical objects, is faithful for any integer $omega eq 1$.
rate research
Read More
We describe all groups that can be generated by two twists along spherical sequences in an enhanced triangulated category. It will be shown that with one exception such a group is isomorphic to an abelian group generated by not more than two elements, the free group on two generators or the braid group of one of the types $A_2$, $B_2$ and $G_2$ factorized by a central subgroup. The last mentioned subgroup can be nontrivial only if some specific linear relation between length and sphericity holds. The mentioned exception can occur when one has two spherical sequences of length $3$ and sphericity $2$. In this case the group generated by the corresponding two spherical twists can be isomorphic to the nontrivial central extension of the symmetric group on three elements by the infinite cyclic group. Also we will apply this result to give a presentation of the derived Picard group of selfinjective algebras of the type $D_4$ with torsion $3$ by generators and relations.
We study Dehn twists along Lagrangian submanifolds that are finite quotients of spheres. We decribe the induced auto-equivalences to the derived Fukaya category and explain its relation to twists along spherical functors.
We study Artin-Tits braid groups $mathbb{B}_W$ of type ADE via the action of $mathbb{B}_W$ on the homotopy category $mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following Brav-Thomas, we define a metric on $mathbb{B}_W$ induced by the canonical $t$-structure on $mathcal{K}$, and prove that this metric on $mathbb{B}_W$ agrees with the word-length metric in the canonical generators of the standard positive monoid $mathbb{B}_W^+$ of the braid group. We also define, for each choice of a Coxeter element $c$ in $W$, a baric structure on $mathcal{K}$. We use these baric structures to define metrics on the braid group, and we identify these metrics with the word-length metrics in the Birman-Ko-Lee/Bessis dual generators of the associated dual positive monoid $mathbb{B}_{W.c}^vee$. As consequences, we give new proofs that the standard and dual positive monoids inject into the group, give linear-algebraic solutions to the membership problem in the standard and dual positive monoids, and provide new proofs of the faithfulness of the action of $mathbb{B}_W$ on $mathcal{K}$. Finally, we use the compatibility of the baric and $t$-structures on $mathcal{K}$ to prove a conjecture of Digne and Gobet regarding the canonical word-length of the dual simple generators of ADE braid groups.
Let $mathfrak{g}$ be a semisimple simply-laced Lie algebra of finite type. Let $mathcal{C}$ be an abelian categorical representation of the quantum group $U_q(mathfrak{g})$ categorifying an integrable representation $V$. The Artin braid group $B$ of $mathfrak{g}$ acts on $D^b(mathcal{C})$ by Rickard complexes, providing a triangulated equivalence $Theta_{w_0}:D^b(mathcal{C}_mu) to D^b(mathcal{C}_{w_0(mu)})$, where $mu$ is a weight of $V$ and $Theta_{w_0}$ is a positive lift of the longest element of the Weyl group. We prove that this equivalence is t-exact up to shift when $V$ is isotypic, generalising a fundamental result of Chuang and Rouquier in the case $mathfrak{g}=mathfrak{sl}_2$. For general $V$, we prove that $Theta_{w_0}$ is a perverse equivalence with respect to a Jordan-Holder filtration of $mathcal{C}$. Using these results we construct, from the action of $B$ on $V$, an action of the cactus group on the crystal of $V$. This recovers the cactus group action on $V$ defined via generalised Schutzenberger involutions, and provides a new connection between categorical representation theory and crystal bases. We also use these results to give new proofs of theorems of Berenstein-Zelevinsky, Rhoades, and Stembridge regarding the action of symmetric group on the Kazhdan-Lusztig basis of its Specht modules.
We compute Witt groups of maximal isotropic Grassmannians, aka. spinor varieties. They are examples of type D homogenuous varieties. Our method relies on the Blow-up setup of Balmer-Calm`es, and we investigate the connecting homomorphism via the projective bundle formula of Walter-Nenashev, the projection formula of Calm`es-Hornbostel and the excess intersection formula of Fasel. The computation in the Type D case can be presented by so called even shifted young diagrams.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
