Majorana Representations of Finite Groups

In this paper we give a brief overview of the theory of Majorana representations of finite groups.     

A method of generating integral representations

Conclusion The objective of this paper was to present a unified method of generating integral representations in special function theory using the idea of simultaneous separability of ²U+U=0 in orthogonal curvilinear coordinate systems. The main results of this paper are formulas (8), (9) and (10) with formula (11) as a special case of (10).This paper contains results submitted by W. W.Turner as a dissertation for the degree of Doctor of Philosophy at Michigan State University.     

A note on square-integrable representations

In this short note, irreducible square-integrable representations of left Hilbert algebras are studied in detail. Orthogonality relations are formulated and proved which contain as a special case the orthogonality relations for square-integrable representations of unimodular locally compact groups. A self-adjoint (generally unbounded) operator is defined on the representation space, the inverse of whose square has some claim to the title “formal dimension operator” since in the case of unimodular groups the inverse of the square of this operator is just the formal dimension times the identity operator. Although the methods used are quite different, this note was inspired by some similar results of M. Duflo and C. Moore for nonunimodular groups.     

Existence of Majorana Bound States in A Superconducting Nanowire Near an Impurity

We consider a nanowire with the s-wave superconducting order induced as a result of the proximity effect in the presence of the Zeeman field and the Rashba interaction. For a small superconducting gap and small momenta, we analytically prove the existence of Majorana bound states for a certain local change in the Zeeman field or the superconducting order and also obtain explicit expressions for the corresponding wave functions. We study the scattering of excited states with energies that are close to boundary gap points in the case of propagation through an impurity for local changes in the indicated system parameters near this impurity and show that the transmission probability is equal to unity.     

Basic spin representations of 2S nand 2A n as globally irreducible representations

Dedicated to Prof. Dr. O. H. Kegel on his 60th birthday     

A bilinear algorithm for sparse representations

We consider the following sparse representation problem: represent a given matrix X∈ℝ ^m×N as a multiplication X=AS of two matrices A∈ℝ ^m×n (m≤n<N) and S∈ℝ ^n×N , under requirements that all m×m submatrices of A are nonsingular, and S is sparse in sense that each column of S has at least n−m+1 zero elements. It is known that under some mild additional assumptions, such representation is unique, up to scaling and permutation of the rows of S. We show that finding A (which is the most difficult part of such representation) can be reduced to a hyperplane clustering problem. We present a bilinear algorithm for such clustering, which is robust to outliers. A computer simulation example is presented showing the robustness of our algorithm.     

A perspective on algebraic representations of lattices

Dedicated to Bjarni Jónsson on his 70th birthday     

A family of affine Weyl group representations

In this paper we explicitly determine the virtual representations of the finite Weyl subgroups of the affine Weyl group on the cohomology of the space of affine flags containing a family of elementsn _t in an affine Lie algebra. We also compute the Euler characteristic of the space of partial flags containingn _t and give a connection with hyperplane arrangements.This paper forms part of my Ph.D. thesis at M.I.T. I was supported by an NSF Graduate Fellowship and NSF grant DMS 9304580.     

A construction of representations of affine Weyl groups

Let G be a complex, semisimple, simply connected algebraic group withLie algebra . We extend scalars to the power series field in one variable C(()), and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of C(()),i.e. fixed point varieties on the full affine flag manifold. We definerepresentations of the affine Weyl group in the homology of these varieties,generalizing Kazhdan and Lusztig's topological construction of Springer'srepresentations to the affine context.     

A class of essential representations of product systems

We construct a family of essential representations of an arbitrary product system by generalizing some techniques introduced by M. Skeide and W. Arveson. We then classify the resulting E₀-semigroups up to conjugacy, by identifying their tail flows as periodic W^∗-dynamical systems acting on factors of type I_∞. The conjugacy classes of these E₀-semigroups correspond to the orbits of the action of the automorphism group of the product system on unital vectors. In the sequel, this classification shows explicitly that any E₀-semigroup admits uncountably many non-conjugate cocycle perturbations.     

A remark on boundary level admissible representations

We point out that it is immediate by our character formula that in the case of a boundary level the characters of admissible representations of affine Kac–Moody algebras and the corresponding W-algebras decompose in products in terms of the Jacobi form ${?}_{11}(\tau ,z)$.