Algorithms for group isomorphism via group extensions and cohomology

The isomorphism problem for finite groups of order n (GpI) has long been known to be solvable in $n^{\log n+O(1)}$ time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups. The extension theory describes how a normal subgroup N is related to G/N via G, and this naturally leads to a divide-and-conquer strategy that splits GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup N is abelian, this strategy is well-known. Our first contribution is to extend this strategy to handle the case when N is not necessarily abelian. This allows us to provide a unified explanation of all recent polynomial-time algorithms for special group classes. Guided by this strategy, to make further progress on GpI, we consider central-radical groups, proposed in Babai et al. (SODA 2011): the class of groups such that G mod its center has no abelian normal subgroups. This class is a natural extension of the group class considered by Babai et al. (ICALP 2012), namely those groups with no abelian normal subgroups. Following the above strategy, we solve GpI in $n^{O(\log \log n)}$ time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GpI in $n^{O(\log\log n)}$ time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time there have been worst-case guarantees on a $n^{o(\log n)}$-time algorithm that tackles both aspects of GpI---actions and cohomology---simultaneously.Comment: 54 pages + 14-page appendix. Significantly improved presentation, with some new result

Tensor product categorifications and the super Kazhdan-Lusztig conjecture

We give a new proof of the "super Kazhdan-Lusztig conjecture" for the Lie super algebra $\mathfrak{gl}_{n|m}(\mathbb{C})$ as formulated originally by the first author. We also prove for the first time that any integral block of category O for $\mathfrak{gl}_{n|m}(\mathbb{C})$ (and also all of its parabolic analogs) possesses a graded version which is Koszul. Our approach depends crucially on an application of the uniqueness of tensor product categorifications established recently by the second two authors.Comment: 58 pages; v2: relatively minor changes, a few adjustments to wording and references; v3: final version, more minor changes, to appear in IMR

N=2 structures on solvable Lie algebras: the c=9 classification

Let G be a finite-dimensional Lie algebra (not necessarily semisimple). It is known that if G is self-dual (that is, if it possesses an invariant metric) then there is a canonical N=1 superconformal algebra associated to its N=1 affinization---that is, it admits an N=1 (affine) Sugawara construction. Under certain additional hypotheses, this N=1 structure admits an N=2 extension. If this is the case, G is said to possess an N=2 structure. It is also known that an N=2 structure on a self-dual Lie algebra G is equivalent to a vector space decomposition G = G_+ \oplus G_- where G_\pm are isotropic Lie subalgebras. In other words, N=2 structures on G are in one-to-one correspondence with Manin triples (G,G_+,G_-). In this paper we exploit this correspondence to obtain a classification of the c=9 N=2 structures on self-dual solvable Lie algebras. In the process we also give some simple proofs for a variety of Lie algebraic results concerning self-dual Lie algebras admitting symplectic or K\"ahler structures.Comment: 49 pages in 2 columns (=25 physical pages), (uufiles-gz-9)'d .dvi file (uses AMSFonts 2.1+). Revision: Added 1 reference, corrected typos, added some more materia

Algebraic Cycles, Fundamental Group of a Punctured Curve, and Applications in Arithmetic

The results of this paper can be divided into two parts, geometric and arithmetic. Let $X$ be a smooth projective curve over $\mathbb{C}$, and $e,\infty\in X(\mathbb{C})$ be distinct points. Let $L_n$ be the mixed Hodge structure of functions on $\pi_1(X-\{\infty\},e)$ given by iterated integrals of length $\leq n$ (as defined by Hain). In the geometric part, inspired by a work of Darmon, Rotger, and Sols, we express the mixed Hodge extension $\mathbb{E}^\infty_{n,e}$ given by the weight filtration on $\frac{L_n}{L_{n-2}}$ in terms of certain null-homologous algebraic cycles on $X^{2n-1}$. As a corollary, we show that the extension $\mathbb{E}^\infty_{n,e}$ determines the point $\infty\in X-\{e\}$. The arithmetic part of the paper gives some number-theoretic applications of the geometric part. We assume that $X=X_0\otimes_K\mathbb{C}$ and $e,\infty\in X_0(K)$, where $K$ is a subfield of $\mathbb{C}$ and $X_0$ is a projective curve over $K$. Let $Jac$ be the Jacobian of $X_0$. We use the extension $\mathbb{E}^\infty_{n,e}$ to associate to each $Z\in CH_{n-1}(X_0^{2n-2})$ a point $P_Z\in Jac(K)$, which can be described analytically in terms of iterated integrals. The proof of $K$-rationality of $P_Z$ uses that the algebraic cycles constructed in the geometric part of the paper are defined over $K$. Assuming a certain plausible hypothesis on the Hodge filtration on $L_n(X-\{\infty\},e)$ holds, we show that an algebraic cycle $Z$ for which $P_Z$ is torsion, gives rise to relations between periods of $L_2(X-\{\infty\},e)$. Interestingly, these relations are non-trivial even when one takes $Z$ to be the diagonal of $X_0$. The geometric result of the paper in $n=2$ case, and the fact that one can associate to $\mathbb{E}^\infty_{2,e}$ a family of points in $Jac(K)$, are due to Darmon, Rotger, and Sols. Our contribution is in generalizing the picture to higher weights.Comment: 65 pages. A few geometric corollaries and an application to periods have been added (compared to the first version