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Majorana representations have been introduced by Ivanov [ Cambridge Tracts in Mathematics , Cambridge University Press, Cambridge, 2009] in order to provide an axiomatic framework for studying the actions on the Griess algebra of the Monster and of its subgroups generated by Fischer involutions. A crucial step in this programme is to obtain an explicit description of the Majorana representations of A_{12} (by Franchi, Ivanov, and Mainardis [J. Algebraic Combin. 44 (2016), pp. 265-292], the largest alternating group admitting a Majorana representation) for this might eventually lead to a new and independent construction of the Monster group (see A.A Ivanov [ Group theory and computation , Indian Stat. Inst. Ser., Springer, Singapore, 2018, Section 4, page 115]). In this paper we prove that A_{12} has two possible Majorana sets, one of which is the set \mathcal X_b of involutions of cycle type 2^2, the other is the union of \mathcal X_b with the set \mathcal X_s of involutions of cycle type 2^6. The latter case (the saturated case) is most interesting, since the Majorana set is precisely the set of involutions of A_{12} that fall into the class of Fischer involutions when A_{12} is embedded in the Monster. We prove that A_{12} has a unique saturated Majorana representation and we determine its degree and decomposition into irreducibles. As consequences we get that the Harada-Norton group has, up to equivalence, a unique Majorana representation and every Majorana algebra, affording either a Majorana representation of the Harada-Norton group or a saturated Majorana representation of A_{12}, satisfies the Straight Flush Conjecture (see A. A. Ivanov [ Contemp. Math. , Amer. Math. Soc., Providence, RI, 2017, pp. 11-17] and A. A. Ivanov [ Group theory and computation , Indian Stat. Inst. Ser., Springer, Singapore, 2018, pp. 107-118]). As a by-product we also determine the degree and the decomposition into irreducibles of the Majorana representation induced on A_8, the four point stabilizer subgroup of A_{12}. We finally state a conjecture about Majorana representations of the alternating groups A_n, 8\leq n\leq 12. [ABSTRACT FROM AUTHOR]

Magnus proved in 1930 that, given two elements x and y of a finitely generated free group F with equal normal closures (x)F = (y)F, x is conjugated either to y or y-1. More recently, this property, called the Magnus property, has been generalized to oriented surface groups. In this paper, we consider an analogue property for profinite surface groups. While the Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for L a class of finite groups, we prove that if x and y are algebraically simple elements of the pro-L completion ...L of an orientable surface group Π such that, for all n ∈ ℕ, there holds (xn)... = (yn)..., then x is conjugated to ys for some s ∈ (...L)*. As a matter of fact, a much more general property is proved and further extended to a wider class of profinite completions. The most important application of the theory above is a generalization of the description of centralizers of profinite Dehn twists given in [Marco Boggi, Trans. Amer. Math. Soc. 366 (2014), 5185-5221] to profinite Dehn multitwists. [ABSTRACT FROM AUTHOR]

Let G be a topological group. Then the based loopspace OG of G is an algebra over the cacti operad, while the double loopspace O2BG of the classifying space of G is an algebra over the framed little discs operad. This paper shows that these two algebras are equivalent, in the sense that they are weakly equivalent E-algebras, where E is an operad weakly equivalent to both framed little discs and cacti. We recover the equivalence between cacti and framed little discs, and Menichi's isomorphism between the BV-algebras H*(OG) and H*(O2BG). [ABSTRACT FROM AUTHOR]

This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on $\mathbb C^n$. A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter groups. The key technical result is that the spherical Deligne complex for the 4-strand braid group is CAT(1). [ABSTRACT FROM AUTHOR]

We give new combinatorial formulas for decomposition of the tensor product of integrable highest weight modules over the classical Lie algebras of types B,C,D, and the branching decomposition of an integrable highest weight module with respect to a maximal Levi subalgebra of type A. This formula is based on a combinatorial model of classical crystals called spinor model. We show that our formulas extend in a bijective way various stable branching rules for classical groups to arbitrary highest weights, including the Littlewood restriction rules. [ABSTRACT FROM AUTHOR]

Let G = exp(g) be a connected, simply connected, nilpotent Lie group and let ω be a continuous symmetric weight on G with polynomial growth. We determine the structure of all the two-sided closed ideals of the weighted group algebra Lω1(G) which are attached to a flat co-adjoint orbit. [ABSTRACT FROM AUTHOR]

We study the automorphism group of a simple, unital, Z-stable C*-algebra. We show that Inn0(A) is a simple topological group and Inn(A)/Inn0(A) is isomorphic (as topological groups) to the inverse limit of quotient groups of K1(A), where A is a Z-stable C*-algebra satisfying the following property: for every UHF algebra B, A ⊗ B is a nuclear, separable, simple, tracially AI algebra satisfying the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet. By the recent results of Lin and Winter, ordered K-theory, traces, and the class of the unit is a complete isomorphism invariant for this class of C*-algebra. [ABSTRACT FROM AUTHOR]

Let C be a unital AH-algebra and let A be a unital separable simple C*-algebra with tracial rank no more than one. Suppose that φ,ψ : C → A are two unital monomorphisms. With some restriction on C, we show that φ and ψ are approximately unitarily equivalent if and only if [φ] = [ψ] in KL(C,A), τ o φ = τ o ψ for all tracial states of A and φ ‡ = ψ ‡ , where φ‡ and ψ‡ are homomorphisms from U(C)/CU(C) → U(A)/CU(A) induced by φ and ψ, respectively, and where CU(C) and CU(A) are closures of the subgroup generated by commutators of the unitary groups of C and B. A more practical but approximate version of the above is also presented. [ABSTRACT FROM AUTHOR]