Space Group Representations. Theory, Tables and Applications. By Nikolai B. Melnikov and Boris I. Reser. Springer Nature Switzerland AG, 2022. ISBN 978-3-031-13990-1.

The book offers a clear explanation of space-group representation theory, focusing on the projective representations of crystallographic point groups and their practical applications in energy band calculations.Detailed tables are provided for constructing irreducible representations of all space groups.The authors aim to promote their approach by making each chapter standalone and accessible for purchase separately.The book targets researchers and graduate students in solid-state physics and chemistry, assuming only a basic understanding of linear algebra and quantum mechanics.However, it is less suitable for crystallographers for a number of reasons that are expanded on below.The book consists of three parts: theory (chapters 2-5), tables (chapter 6) and applications (chapters 7-17).Within the appendices, the authors have assembled proofs of certain mathematical theorems, along with supplementary content on projective representations and double-valued representations.The initial part of the book consists of an introductory chapter followed by three central chapters.Chapter 2 gives a concise overview of group theory, linear algebra and representation theory.Chapter 3 focuses on induced representations, while chapter 4 covers projective representations, which, according to the method developed in the book, are essential for the construction of space-group representations.To ensure self-sufficiency, chapters 3 and 4 provide proofs for most assertions and theorems in a constructive and concise manner.Practitioners seeking immediate applications may choose to defer or revisit these chapters later as they require some level of abstraction.In chapter 5 the theoretical results of the previous chapters are applied to the construction of irreducible representations of space groups.Specifically, the focus is on irreducible projective representations of the 32 crystallographic point groups, with a convenient approach for deriving them.As a next step, the matrices of all irreducible (single-and double-valued) representations of a space group are derived from the irreducible projective representations of the 32 crystallographic point groups.Additionally, the text discusses the role of spin and the consequences of time-reversal symmetry.Part II features chapter 6, a crucial component of the book with comprehensive tables of irreducible representations for space groups.This chapter is substantial in both content and size and serves as a standalone reference.Section 6.1 gives a brief overview of the preceding theory and offers guidance on using the tables.Sections 6.2-6.19 contain condensed tables displaying irreducible projective representations of the 32 crystallographic point groups, which form the basis for deriving all irreducible representations, including double-valued ones, for all 230 space groups.Section 6.20 provides practical methods for using these tables.Part III focuses on applying group theory in solid-state physics and energy band calculations, and is divided into four sections based on application areas.The first section (covering chapters 7-9) includes the exploration of symmetry in classifying quantummechanical eigenstates, followed by the analysis of its application to solid-state physics, considering also spin and time-reversal symmetry.Chapter 9 introduces electronic energy band calculations, emphasizing symmetry in the orthogonal plane wave (OPW) method.To illustrate representation theory practically, the second section (chapters 10-12) offers real-world examples, especially focusing on non-symmorphic space groups.