Unitary Vector Spaces Research Articles

Supercharacters of the Sylowp-subgroups of the finite symplectic and orthogonal groups

We define and study supercharacters of the classical finite unipotent groups of types Bn(q), Cn(q) and Dn(q). We show that the results we proved in 2006 remain valid over any finite field of odd characteristic. In particular, we show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. In addition, we prove that the unitary vector space spanned by all the supercharacters is closed under multiplication, and establish a formula for the supercharacter values. As a consequence, we obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we give a combinatorial description of all the irreducible characters of maximum degree in terms of the root system, by showing how they can be obtained as constituents of particular supercharacters.

Strongly regular graphs defined by spreads

Spreads of finite symplectic, orthogonal and unitary vector spaces are used to construct new strongly regular graphs having the same parameters as the perpendicularity graphs of the underlying vector spaces. Some of the graphs are related to partial geometries, while others produce interesting symmetric designs.

Inequalities for some Monotone Matrix Functions

Let V denote a unitary vector space with inner product (x, y). A self-adjoint linear map T: V → V is positive (positive definite) if (Tx, x) ≧ 0 ((Tx, x) ≧ 0) for all x ≠ 0 in V. We write S ≧ T(S > T) if S and T are self-adjoint and S – T ≧ 0 (S – T > 0). If U is a unitary vector space, a function f: Hom(V, V) → Hom(U, U) is monotone idf S ≧ T implies that f(S) ≧ f(T). If both U and V are taken to be the n-dimensional unitary space Cn of n-tuples of complex numbers with standard inner product, then f is a monotone matrix junction, a notion introduced for a more restrictive class of functions by Löwner (3) which has important applications in pure and applied mathematics. For orientation we refer the reader to (1), where several interesting examples of monotone and related functions are displayed in detail.

Foundations for the Theory of Least Squares

Summary The Bayesian theory of least squares is founded upon a weaker and more tangible form of prior knowledge than the conventional assumption of normality. The underlying assumption is a form of conditional uniformity on spheres for the “actual errors” in the experiment. This provides a unified theory appropriate for randomization models in the analysis of variance as well as for classical least-squares analysis.

Communication Relations of Quantum Mechanics

The mathematical and physical meaning of the commutation relations of nonrelativistic quantum mechanics is discussed in terms of the representation of translations, Galilean transformations, and rotations of the coordinate system by unitary transformations acting on the unitary vector space of quantum states.