Linear sections of GL(4, 2)

For V = V (n; q); a linear section of GL(V ) = GL(n; q) is a vector subspace S of the n 2 -dimensional vector space End(V ) which is contained in GL(V ) [ f0g: We pose the problem, for given (n; q); of classifying the di erent kinds of maximal linear sections of GL(n; q): If S is any linear section of GL(n; q) then dim S n: The case of GL(4; 2) is examined fully. Up to a suitable notion of equiv- alence there are just two classes of 3-dimensional maximal normalized linear sections M3;M0 3 , and three classes M4;M0 4 ;M00 4 of 4-dimensional sections. The subgroups of GL(4; 2) generated by representatives of these ve classes are respectively G3 = A7; G 0 3 = GL(4; 2); G4 = Z15; G 0 4 = Z3 A5; G 00 4 = GL(4; 2): On various occasions use is made of an isomorphism T : A8 ! GL(4; 2): In particular a representative of the class M3 is the image under T of a subset f1; ::: ; 7g of A7 with the property that 1 i j is of order 6 for all i =6 j: The classes M3;M0 3 give rise to two classes of maximal partial spreads of order 9 in PG(7; 2); and the classes M0 4 ;M00 4 yield the two isomorphism classes of proper semi eld planes of order 16

Beyond E11

We study the non-linear realisation of E11 originally proposed by West with particular emphasis on the issue of linearised gauge invariance. Our analysis shows even at low levels that the conjectured equations can only be invariant under local gauge transformations if a certain section condition that has appeared in a different context in the E11 literature is satisfied. This section condition also generalises the one known from exceptional field theory. Even with the section condition, the E11 duality equation for gravity is known to miss the trace component of the spin connection. We propose an extended scheme based on an infinite-dimensional Lie superalgebra, called the tensor hierarchy algebra, that incorporates the section condition and resolves the above issue. The tensor hierarchy algebra defines a generalised differential complex, which provides a systematic description of gauge invariance and Bianchi identities. It furthermore provides an E11 representation for the field strengths, for which we define a twisted first order self-duality equation underlying the dynamics.Comment: 97 pages. v2: Minor changes, references added. Published versio

A pedagogical presentation of a C⋆C^\star-algebraic approach to quantum tomography

It is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schrodinger-Dirac picture of quantum mechanics on Hilbert spaces. In this picture states are a primary concept and observables are derived from them. On the other hand, the Heisenberg picture,which has evolved in the $C^\star-$algebraic approach to quantum mechanics, starts with the algebra of observables and introduce states as a derived concept. The equivalence between these two pictures amounts essentially, to the Gelfand-Naimark-Segal construction. In this construction, the abstract $C^\star-$algebra is realized as an algebra of operators acting on a constructed Hilbert space. The representation one defines may be reducible or irreducible, but in either case it allows to identify an unitary group associated with the $C^\star-$algebra by means of its invertible elements. In this picture both states and observables are appropriate functions on the group, it follows that also quantum tomograms are strictly related with appropriate functions (positive-type)on the group. In this paper we present, by means of very simple examples, the tomographic description emerging from the set of ideas connected with the $C^\star-$algebra picture of quantum mechanics. In particular, the tomographic probability distributions are introduced for finite and compact groups and an autonomous criterion to recognize a given probability distribution as a tomogram of quantum state is formulated

On the E10/Massive Type IIA Supergravity Correspondence

In this paper we investigate in detail the correspondence between E10 and Romans' massive deformation of type IIA supergravity. We analyse the dynamics of a non-linear sigma model for a spinning particle on the coset space E10/K(E10) and show that it reproduces the dynamics of the bosonic as well as the fermionic sector of the massive IIA theory, within the standard truncation. The mass deformation parameter corresponds to a generator of E10 outside the realm of the generators entering the usual D=11 analysis, and is naturally included without any deformation of the coset model for E10/K(E10). Our analysis thus provides a dynamical unification of the massless and massive versions of type IIA supergravity inside E10. We discuss a number of additional and general features of relevance in the analysis of any deformed supergravity in the correspondence to Kac-Moody algebras, including recently studied deformations where the trombone symmetry is gauged.Comment: 68 pages, including 5 appendices, 5 figures. v2: Typos corrected, published version. v3:Title correcte

On the Gelfand property for complex symmetric pairs

We first prove, for pairs consisting of a simply connected complex reductive group together with a connected subgroup, the equivalence between two different notions of Gelfand pairs. This partially answers a question posed by Gross, and allows us to use a criterion due to Aizenbud and Gourevitch, and based on Gelfand-Kazhdan's theorem, to study the Gelfand property for complex symmetric pairs. This criterion relies on the regularity of the pair and its descendants. We introduce the concept of a pleasant pair, as a means to prove regularity, and study, by recalling the classification theorem, the pleasantness of all complex symmetric pairs. On the other hand, we prove a method to compute all the descendants of a complex symmetric pair by using the extended Satake diagram, which we apply to all pairs. Finally, as an application, we prove that eight out of the twelve exceptional complex symmetric pairs, together with the infinite family $(\textrm{Spin}_{4q+2}, \textrm{Spin}_{4q+1})$, satisfy the Gelfand property, and state, in terms of the regularity of certain symmetric pairs, a sufficient condition for a conjecture by van Dijk and a reduction of a conjecture by Aizenbud and Gourevitch.Comment: Presentation and arguments improved in Sections 5.1 and 5.2. Typos and small mistakes fixe

Large subgroups of simple groups

Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this note we determine all the large maximal subgroups of finite simple groups, and we establish an analogous result for simple algebraic groups (in this context, largeness is defined in terms of dimension). An application to triple factorisations of simple groups (both finite and algebraic) is discussed.Comment: 37 page

Products of conjugacy classes and fixed point spaces

We prove several results on products of conjugacy classes in finite simple groups. The first result is that there always exists a uniform generating triple. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth powers for any prime power r (in particular, a product of two squares).Comment: 44 page