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The Freudenthal magic square is a specific arrangement of numbers that forms a 3x3 grid where the sums of the numbers in each row, column, and the two main diagonals all equal the same value, thus giving it the properties of a magic square. It is named after the Dutch mathematician Hans Freudenthal.

The Gelfand–Graev representation is a specific type of representation associated with the theory of finite groups, particularly in the context of group algebras and representation theory. Named after I. M. Gelfand and M. I. Graev, this representation is a construction that arises in the study of group characters and modular representations.

A glossary of representation theory typically includes definitions and explanations of key terms and concepts used in the field of representation theory, which is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces.

The Maass–Selberg relations are a set of identities that relate certain arithmetic functions associated with modular forms and automorphic forms to equivalent forms involving Dirichlet series and other number-theoretic objects. They were developed in the context of the study of modular forms, particularly by mathematicians Hans Maass and Atle Selberg.

David T. Hon is an entrepreneur and inventor known for his work in various technology-related fields, including telecommunications, manufacturing, and medical devices. He is particularly recognized for his contributions to the development of innovative products and materials. Hon holds several patents and has been involved in the creation of companies focused on advanced materials and high-tech consumer products.

Conjunctive grammar is a formal grammar framework that extends traditional context-free grammars, primarily used in the field of computational linguistics and formal language theory. In conjunctive grammar, the productions (rules) allow the combination of multiple rules and conditions to generate strings in a more complex way than simple context-free grammars. The key feature of conjunctive grammars is their use of conjunctions in the grammar rules.

A Hopf algebra is an algebraic structure that is equipped with both algebra and coalgebra structures, together with a certain compatibility condition between them. It is a fundamental concept in abstract algebra, representation theory, and category theory.

The Iwahori–Hecke algebra is a mathematical structure that arises in the study of representation theory, particularly in the representation theory of the symmetric group and related algebraic objects, such as Coxeter groups and reductive algebraic groups. ### Definition The Iwahori–Hecke algebra, often denoted as \( \mathcal{H} \), is an algebra associated with a Coxeter group.

Jantzen filtration is a concept in the field of representation theory, specifically in the study of semisimple Lie algebras and their representations. The filtration is named after Jan Jantzen, who made significant contributions to this area of mathematics.

The Langlands–Shahidi method is a technique in number theory and the theory of automorphic forms that provides a way to study L-functions and their special values, particularly through the lens of the Langlands program. This method is named after two mathematicians: Robert Langlands and Freydoon Shahidi, who have made significant contributions to this area of mathematical research.

Nil-Coxeter algebras are a specific type of algebraic structure that arises in the study of Coxeter systems, particularly in relation to their representations and combinatorial properties. The term generally refers to the algebra associated with a Coxeter group in which the relations are more relaxed, allowing for nilpotent behavior.

Schur's lemma is a fundamental result in representation theory, particularly in the context of representation of groups and algebras. It applies to representations of a group and its modules over a division ring or field.

Paraphrasing is the process of rewording or restating a piece of text or speech while preserving its original meaning. It involves altering the structure, vocabulary, and phrasing of the content without changing its essential message. Paraphrasing can be useful for clarifying information, avoiding plagiarism, or tailoring content for different audiences. It typically requires a good understanding of the original material to accurately convey the same ideas in a new way.

Parechesis is a rhetorical term used to describe a figure of speech in which a word is used in a way that it is not intended to refer to its literal meaning, often for the sake of introducing ambiguity or engaging an audience. It typically involves a form of pun or wordplay, where a word has more than one meaning or can be interpreted in multiple ways.

Parrhesia is a term that originates from ancient Greek, meaning "free speech" or "boldness of speech." It refers to the act of speaking candidly and openly, often about important or controversial topics, without fear of the consequences. The concept is closely associated with the idea of truth-telling and moral courage, where individuals express their thoughts and opinions honestly, even when it might be uncomfortable or risky to do so.

A "pericope" is a term used primarily in biblical studies and literature to refer to a specific section or excerpt of a text, particularly from the Bible. The word comes from the Greek "perikopē," which means "a cutting out" or "a section." In the context of biblical studies, a pericope usually refers to a story, parable, or teaching that is read and interpreted as a distinct unit within Scripture.

American logicians refer to philosophers and scholars in the United States who focus on the study of logic, a branch of philosophy and mathematics that deals with the principles of valid reasoning and inference. The history of American logic is rich and varied, with key figures contributing to different areas of the field.

"American mathematician stubs" refers to short or incomplete articles on American mathematicians that have been created on platforms like Wikipedia. These stub articles typically provide only basic information, such as the mathematician's name, a brief mention of their contributions, and perhaps a few personal details, but lack comprehensive coverage of their work, achievements, and impact on the field of mathematics.

A prehomogeneous vector space is a concept from the field of invariant theory and representation theory, particularly concerning vector spaces that admit a group action with certain properties.

A reductive dual pair is a concept that arises in the context of representation theory and Lie groups. Specifically, it refers to a pair of reductive algebraic groups (or Lie groups) that have compatible structures allowing for the decomposition of representations in a certain way. The term is primarily used in the study of harmonic analysis on groups and has implications in various fields, including number theory, geometry, and mathematical physics. ### Key Points 1.