Induced Representation Method in the Theory of Electron Structure and Superconductivity

It is shown that the application of theorems of induced representations method, namely, Frobenius reciprocity theorem, transitivity of induction theorem, and Mackey theorem on symmetrized squares, makes simplifying standard techniques in the theory of electron structure and constructing Cooper pair wavefunctions on the basis of one-electron solid-state wavefunctions possible. It is proved that the nodal structure of topological superconductors in the case of multidimensional irreducible representations is defined by additional quantum numbers. The technique is extended on projective representations in the case of nonsymmorphic space groups and examples of applications for topological superconductors UPt3 and Sr2RuO4 are considered.

A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM

Let $G$ be a locally compact group and $K$ a closed subgroup of $G$. Let $\unicode[STIX]{x1D6FE},$$\unicode[STIX]{x1D70B}$ be representations of $K$ and $G$ respectively. Moore’s version of the Frobenius reciprocity theorem was established under the strong conditions that the underlying homogeneous space $G/K$ possesses a right-invariant measure and the representation space $H(\unicode[STIX]{x1D6FE})$ of the representation $\unicode[STIX]{x1D6FE}$ of $K$ is a Hilbert space. Here, the theorem is proved in a more general setting assuming only the existence of a quasi-invariant measure on $G/K$ and that the representation spaces $\mathfrak{B}(\unicode[STIX]{x1D6FE})$ and $\mathfrak{B}(\unicode[STIX]{x1D70B})$ are Banach spaces with $\mathfrak{B}(\unicode[STIX]{x1D70B})$ being reflexive. This result was originally established by Kleppner but the version of the proof given here is simpler and more transparent.

Principal bundles as Frobenius adjunctions with application to geometric morphisms

Using a suitable notion of principalG-bundle, defined relative to an arbitrary cartesian category, it is shown that principal bundles can be characterised as adjunctions that stably satisfy Frobenius reciprocity. The result extends from internal groups to internal groupoids. Since geometric morphisms can be described as certain adjunctions that are stably Frobenius, as an application it is proved that all geometric morphisms, from a localic topos to a bounded topos, can be characterised as principal bundles.

POSITIVE REPRESENTATIONS OF FINITE GROUPS IN RIESZ SPACES

In this paper, which is part of a study of positive representations of locally compact groups in Banach lattices, we initiate the theory of positive representations of finite groups in Riesz spaces. If such a representation has only the zero subspace and possibly the space itself as invariant principal bands, then the space is Archimedean and finite-dimensional. Various notions of irreducibility of a positive representation are introduced and, for a finite group acting positively in a space with sufficiently many projections, these are shown to be equal. We describe the finite-dimensional positive Archimedean representations of a finite group and establish that, up to order equivalence, these are order direct sums, with unique multiplicities, of the order indecomposable positive representations naturally associated with transitive G-spaces. Character theory is shown to break down for positive representations. Induction and systems of imprimitivity are introduced in an ordered context, where the multiplicity formulation of Frobenius reciprocity turns out not to hold.

RETRACTIVE TRANSFERS AND p-LOCAL FINITE GROUPS

In this paper we explore the possibility of defining $p$-local finite groups in terms of transfer properties of their classifying spaces. More precisely, we consider the question, posed by Haynes Miller, of whether an equivalent theory can be obtained by studying triples $(f,t,X)$, where $X$ is a $p$-complete, nilpotent space with a finite fundamental group, $f:BS\to X$ is a map from the classifying space of a finite $p$-group, and $t$ is a stable retraction of $f$ satisfying Frobenius reciprocity at the level of stable homotopy. We refer to $t$ as a retractive transfer of $f$ and to $(f,t,X)$ as a retractive transfer triple over $S$.In the case where $S$ is elementary abelian, we answer this question in the affirmative by showing that a retractive transfer triple $(f,t,X)$ over $S$ does indeed induce a $p$-local finite group over $S$ with $X$ as its classifying space.Using previous results obtained by the author, we show that the converse is true for general finite $p$-groups. That is, for a $p$-local finite group $(S,\mathcal{F},\mathcal{L})$, the natural inclusion $\theta:BS\to X$ has a retractive transfer $t$, making $(\theta,t,|\mathcal{L}|^{\wedge}_p)$ a retractive transfer triple over $S$. This also requires a proof, obtained jointly with Ran Levi, that $|\mathcal{L}|^{\wedge}_p$ is a nilpotent space, which is of independent interest.