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Register a new userOn the 2-linearity of the free group
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Authors
Anthony M. Licata
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We construct an action of the free group $F_n$ on the homotopy category of projective modules over a finite dimensional zigzag algebra. The main theorem in the paper is that this action is faithful. We describe the relationship between homotopy classes of paths in the punctured disc and complexes of projective zigzag modules and explore the connection between gradings on the zigzag algebra and monoids in $F_n$. We use this connection to give homological constructions of the standard and Bessis dual word length metrics on the free group.
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We prove that if $B$ is a $p$-block with non-trivial defect group $D$ of a finite $p$-solvable group $G$, then $ell(B) < p^r$, where $r$ is the sectional rank of $D$. We remark that there are infinitely many $p$-blocks $B$ with non-Abelian defect groups and $ell(B) = p^r - 1$. We conjecture that the inequality $ell(B) leq p^r$ holds for an arbitrary $p$-block with defect group of sectional rank $r$. We show this to hold for a large class of $p$-blocks of various families of quasi-simple and nearly simple groups.
The Hochschild cohomology ring of a group algebra is an object that has received recent attention, but is difficult to compute, in even the simplest of cases. In this paper, we use the product formula due to Witherspoon and Siegel to extend some of their computations. In particular, we compute the Hochschild cohomology algebra of group algebras kG where |G| is less than 16, and we provide an alternative computation of the ring $HH^*(k(E ltimes P))$ considered by Kessar and Linckelmann.
For $4 mid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level $L$ subgroup of the mapping class group (in a previous paper, the author determined this rationally). This entails calculating the abelianization of the level $L$ subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally, along the way we calculate the first homology group of the mod $L$ symplectic group with coefficients in the adjoint representation.
Let $U$ be a Sylow $p$-subgroup of the finite Chevalley group of type $D_4$ over the field of $q$ elements, where $q$ is a power of a prime $p$. We describe a construction of the generic character table of $U$.
Any group $G$ gives rise to a 2-group of inner automorphisms, $mathrm{INN}(G)$. It is an old result by Segal that the nerve of this is the universal $G$-bundle. We discuss that, similarly, for every 2-group $G_{(2)}$ there is a 3-group $mathrm{INN}(G_{(2)})$ and a slightly smaller 3-group $mathrm{INN}_0(G_{(2)})$ of inner automorphisms. We describe these for $G_{(2)}$ any strict 2-group, discuss how $mathrm{INN}_0(G_{(2)})$ can be understood as arising from the mapping cone of the identity on $G_{(2)}$ and show that its underlying 2-groupoid structure fits into a short exact sequence $G_{(2)} to mathrm{INN}_0(G_{(2)}) to Sigma G_{(2)}$. As a consequence, $mathrm{INN}_0(G_{(2)})$ encodes the properties of the universal $G_{(2)}$ 2-bundle.
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