Gamma homology,Lie representations and <Emphasis Type="Italic">E</Emphasis><Subscript>∞</Subscript> multiplications

We prove that the stable homotopy of any Γ-module F is the homology of a bicomplex Ξ(F), in which the (q−1)st row is the two-sided bar construction ℬ(Lie^* _q ,Σ _q ,F[q]). This gives a natural homotopical cotangent bicomplex for graded commutative algebras, in a form suitable for use in a new obstruction theory for classifying E _∞ ring structures on spectra. The E _∞ structure on certain Lubin-Tate spectra is a corollary. Oblatum 15-X-2001 & 14-X-2002?Published online: 24 February 2003     

The spectrum of the Dirac operator on SU<Subscript>2</Subscript>/Q<Subscript>8</Subscript>

We compute the fundamental Dirac operator for the three-parameter-family of homogeneous Riemannian metrics and the four different spin structures on SU₂/Q₈, where Q₈ denotes the group of quaternions. We deduce its spectrum for the Berger metrics and show the sharpness of Christian Bär’s upper bound for the smallest Dirac eigenvalue in the particular case where SU₂/Q₈ is a homogeneous minimal hypersurface of S ⁴.     

Recognition of the groups <Emphasis Type="Italic">E</Emphasis><Subscript>7</Subscript>(2) and <Emphasis Type="Italic">E</Emphasis><Subscript>7</Subscript>(3) by prime graph

It is proved that the groups E ₇(2) and E ₇(3) are recognizable by their prime graphs. As a corollary, this completes the proof of V.D. Mazurov’s conjecture that every finite simple group whose prime graph has at least three connected components is either recognizable by spectrum or isomorphic to A ₆.     

A New Method for Finding the Characteristic Roots of E<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>/E<Subscript><Emphasis Type="Italic">m</Emphasis></Subscript>/1 Queues

In 1953, Smith (Proc Camb Philos Soc 49:449–461, 1953), and, following him, Syski (1960) suggested a method to find the waiting time distribution for one server queues with Erlang-n arrivals and Erlang-m service times by using characteristic roots. Syski shows that these roots can be determined from a very simple equation, but an equation of degree n + m. Syski also shows that almost all of the characteristic roots are complex. In this paper, we derive a set of equations, one for each complex root, which can be solved by Newton’s method using real arithmetic. This method simplifies the programming logic because it avoids deflation and the subsequent polishing of the roots. Using the waiting time distribution, Syski then derived the distribution of the number in the system after a departure. E _n /E _m /1 queues can also formulated as quasi birth-death (QBD) processes, and in this case, the characteristic roots discussed by Syski are closely related to the eigenvalues of the QBD process. The QBD process provides information about the number in system at random times, but they are much more difficult to formulate and solve.     

Groups of outer type E<Subscript>6</Subscript> with trivial Tits algebras

In two 1966 papers, J. Tits gave a construction of exceptional Lie algebras (hence implicitly exceptional algebraic groups) and a classification of possible indexes of simple algebraic groups. For the special case of his construction that gives groups of type E₆, we connect the two papers by answering the question: Given an Albert algebra A and a separable quadratic field extension K, what is the index of the resulting algebraic group?     

Explicit computations of low-lying eigenfunctions for the quantum trigonometric Calogero-Sutherland model related to the exceptional algebra <Emphasis Type="Italic">E</Emphasis><Subscript>7</Subscript>

In a previous paper, we studied the characters and Clebsch-Gordan series for the exceptional Lie algebra E₇ by relating them to the quantum trigonometric Calogero-Sutherland Hamiltonian with the coupling constant κ = 1. We now extend that approach to the case of an arbitrary coupling constant. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 283–293, February, 2008.     

Chevalley groups of type E<Subscript>6</Subscript> in the 27-dimensional representation

The present paper is devoted to a detailed computer study of the action of the Chevalley group G(E ₆, R) on the minimal module V(ῶ ₁). Our main objectives are an explicit choice and a tabulation of the signs of structure constants for this action, compatible with the choice of a positive Chevalley base, the construction of multilinear invariants and equations on the matrix entries of matrices from G(E ₆, R) in this representation, and an explicit tabulation of root elements. Bibliography: 170 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 5–68.     

A new functor from <Emphasis Type="Italic">D</Emphasis><Subscript>5</Subscript>-Mod to <Emphasis Type="Italic">E</Emphasis><Subscript>6</Subscript>-Mod

We find a new representation of the simple Lie algebra of type E ₆ on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen’s idea of mixed product, we construct a new functor from D ₅-Mod to E ₆-Mod. A condition for the functor to map a finite-dimensional irreducible D ₅-module to an infinite-dimensional irreducible E ₆-module is obtained. Our results yield explicit constructions of certain infinite-dimensional irreducible weight E6-modules with finite-dimensional weight subspaces. In our approach, the idea of Kostant’s characteristic identities plays a key role.     

New equations of infinitesimal deformations of surfaces in <Emphasis Type="Italic">E</Emphasis><Subscript>3</Subscript>

We establish a condition for two symmetric tensor fields that is necessary and sufficient for the existence of a displacement vector in the case of infinitesimal deformation of a surface in the Euclidean space E ₃.     

Non-triviality of a product in the Adams <Emphasis Type="Italic">E</Emphasis> <Subscript>2</Subscript>-term

Let p be a prime greater than five and A the mod p Steenrod algebra. In this paper, we prove that \(h_n h_m \tilde \delta _{s + 4} \in Ext_A^{s + 6,t(s,n,m) + s} (Z/p,Z/p)\) is nontrivial in the Adams E₂-term when m ≥ n + 2 ≥ 7 and 0 ≤ s < p ? 4, and trivial in the Adams E₂-term when m ≥ n + 2 = 6 and 0 ≤ s < p ? 4, where \(\tilde \delta _{s + 4} \) stands for the fourth Greek letter element and t(s, n, m) = 2(p ? 1)[(s + 1) + (s + 2)p + (s + 3)p² + (s + 4)p³ + pⁿ + p^m].     

Towards the Hurwitz Generation of G<Subscript>sc</Subscript>(E<Subscript>6</Subscript>, <Emphasis Type="Italic">q</Emphasis>)

The paper is devoted to the problem concerning the Hurwitz generation of the group G_sc(E₆, q). All possibilities for Hurwitz generators, except for just one, are excluded. Bibliography: 25 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 305, 2003, pp. 228–237.     

<Emphasis Type="Italic">D</Emphasis><Subscript>∞</Subscript>-differential <Emphasis Type="Italic">E</Emphasis><Subscript>∞</Subscript>-algebras and Steenrod operations in spectral sequences

This paper is devoted to the introduction of a D _∞-differential analog of the notion of an E _∞-(co)algebra and to the construction of generalized Steenrod operations in terms of multiplicative spectral sequences. In this paper, we investigate basic homotopy properties of D _∞-differential E _∞-(co)algebras and construct a spectral sequence of a D _∞-differential E _∞-(co)algebra. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.     

The E<Subscript>t</Subscript>-Construction for Lattices,Spheres and Polytopes

We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction does not always specialize to convex polytopes; however, in a number of cases where we can realize it, it produces interesting classes of polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple 4-polytopes, as requested by Eppstein et al. We also construct for each d 3 an infinite family of (d – 2)-simplicial 2-simple d-polytopes, thus solving a problem of Grünbaum.     

Geometric cycles with local coefficients and the cohomology of arithmetic subgroups of the exceptional group <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

We study the cohomology of a compact locally symmetric space attached to an arithmetic subgroup of a rational form of a group of type G ₂ with values in a finite dimensional irreducible representation E of G ₂. By constructing suitable geometric cycles and parallel sections of the bundle [(E)\tilde]{\tilde{E}} we prove non-vanishing results for this cohomology. We prove all possible non-vanishing results compatible with the known vanishing theorems regarding unitary representations with non-zero cohomology in the case of the short fundamental weight of G ₂. A decisive tool in our approach is a formula for the intersection numbers with local coefficients of two geometric cycles.     

Coxeter group actions on <Subscript>4</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>3</Subscript>(1) hypergeometric series

In this paper we investigate a certain linear combination K([(x)\vec])=K(a;b,c,d;e,f,g)K(\vec{x})=K(a;b,c,d;e,f,g) of two Saalschutzian hypergeometric series of type ₄ F ₃(1). We first show that K([(x)\vec])K(\vec{x}) is invariant under the action of a certain matrix group G _K , isomorphic to the symmetric group S ₆, acting on the affine hyperplane V={(a,b,c,d,e,f,g)∈ℂ⁷:e+f+g−a−b−c−d=1}. We further develop an algebra of three-term relations for K(a;b,c,d;e,f,g). We show that, for any three elements μ ₁,μ ₂,μ ₃ of a certain matrix group M _K , isomorphic to the Coxeter group W(D ₆) (of order 23040) and containing the above group G _K , there is a relation among K(m₁[(x)\vec])K(\mu_{1}\vec{x}), K(m₂[(x)\vec])K(\mu_{2}\vec{x}), and K(m₃[(x)\vec])K(\mu_{3}\vec{x}), provided that no two of the μ _j ’s are in the same right coset of G _K in M _K . The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in a,b,c,d,e,f,g.     

A Steady-State Approximation for the E<Superscript>x</Superscript><Subscript>k</Subscript><Emphasis Type="Italic">/M/</Emphasis>1 Queue

A simple approximate formula for the steady-state average number of customers in the E^x_k/M/1 queueing system is derived and evaluated. The formula is very accurate: the errors incurred are small and insensitive to the shape of the batch-size distribution and to the values of the parameters involved; in the special case where the batch size has a geometric distribution, the formula appears to be exact.     

P-Adic analysis and the transfinite E8 exceptional Lie symmetry group unification

In P-Adic analysis like in a fractal Cantorian space there is no absolute scale. P-Adic analysis with its prime numbers base is the mathematical quarks of the exceptional E8 and E-infinity. The P-Adic space permits the use of Weyl original spacetime gauge theory which is the rationale behind E-infinity.