2016 Fiscal Year Annual Research Report
Cohomology of Artin groups
| Project/Area Number |
16J00125
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| Research Institution | Hokkaido University |
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Principal Investigator |
劉 曄 北海道大学, 大学院理学院, 特別研究員(DC2)
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| Project Period (FY) |
2016-04-22 – 2018-03-31
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| Keywords | Artin groups / Coxeter groups / Group homology |
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| Outline of Annual Research Achievements |
1. In joint work with T. Akita, we have computed the second mod 2 homology of arbitrary Artin groups by using combinatorial group theoretic techniques. Our result states that this homology is read off from the associated Coxeter graph of the Artin group. Corollaries on stability of second mod 2 homology of certain families of Artin groups are also obtained. Unlike previous researches, this result does not assume that the $K(\pi,1)$ conjecture is true. This also provides affirmative evidence for the conjecture.
2. By studying chromatic functors of graphs introduced by M. Yoshinaga, we obtained the structure of the automorphism group of the chromatic functor, in terms of the combinatorial data of that graph, to be specific, the stable partition. Furthermore, we have found a new example of representation stability using chromatic functors. That is, we consider the composition of the chromatic functor of a finite graph, restricted on the category $\mathrm{FI}$ of finite sets and injections, with the free functor taking a finite set $S$ to the complex vector space with basis $S$. What we proved is that this composition functor is a finitely generated $\mathrm{FI}$-module.
3. In joint work with T. Akita, we have generalized Kleshchev-Nakano and Burichenko's vanishing result of mod p cohomology of alternating groups to that of alternating subgroups of Coxeter groups, making use of the topology of the Coxeter complex.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
1. Our formula for the second mod 2 homology of arbitrary Artin groups seems to be the first non-trivial result on the homology of arbitrary Artin groups. Existing results of (co)homology of Artin groups are all specific computations for certain type of Artin groups for which the $K(\pi,1)$ conjecture has been proved. Our strategy allows us to avoid the obstacle, and hence is a new direction to the subject.
2. Representation stability is a relatively new research topic and proved useful, for instance, in proving (co)homological stability for Artin groups. Our new example relates it with a combinatorial setting.
3. Our vanishing result of mod p cohomology of alternating subgroups of Coxeter groups is a successful follow-up of T. Akita's vanishing result of p-local homology of Coxeter groups.
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| Strategy for Future Research Activity |
1. Future research on (co)homology of Artin groups contains two directions. (A) To compute low dimensional (co)homology of Artin groups with various coefficients. (B) To compute all (c)homology of a certain type of Artin groups for which the $K(\pi,1)$ conjecture holds. For (A), we may further our methods using combinatorial group theory, such as Hopf's formula for higher dimensional homology of a group. For (B), we may keep considering affine type C Artin groups.
2. Although for affine type Coxeter groups, the cohomology (over complex) of complement to the affine Coxeter arrangement is infinite dimensional. When regarded as a representation of the corresponding Coxeter group, it is interesting to look for a stability phenomenon.
3. $K(\pi,1)$ conjecture for certain type Artin groups via computation of homology of Salvetti complex with group ring coefficients.
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