Algebra Books

Book SynopsisThe aim of this book is to give an introduction to what has come to be known as Standard Monomial Theory (SMT). SMT deals with the construction of nice bases of finite dimensional irreducible representations of semi-simple algebraic groups or, in geometric terms, nice bases of coordinate rings of flag varieties (and their Schubert subvarieties) associated to these groups. Besides its intrinsic interest, SMT has applications to the study of the geometry of Schubert varieties. SMT has its origin in the work of Hodge, giving bases of the coordinate rings of the Grassmannian and its Schubert subvarieties by ""standard monomials"". In its modern form, SMT was developed by the author in a series of papers written in collaboration with V. Lakshmibai and C. Musili.This book is a reproduction of a course of lectures given by the author in 1983-84 which appeared in the Brandeis Lecture Notes series. The aim of this course was to give an introduction to the series of papers by concentrating on the case of the full linear group. In recent years, there has been great progress in SMT due to the work of Peter Littelmann. Seshadri's course of lectures (reproduced in this book) remains an excellent introduction to SMT.In the second edition, Conjectures of a Standard Monomial Theory (SMT) for a general semi-simple (simply-connected) algebraic group, due to Lakshmibai, have been added as Appendix C. Many typographical errors have been corrected, and the bibliography has been revised.

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Book SynopsisThis book is a volume of the Springer Briefs in Mathematical Physics and serves as an introductory textbook on the theory of Macdonald polynomials. It is based on a series of online lectures given by the author at the Royal Institute of Technology (KTH), Stockholm, in February and March 2021. Macdonald polynomials are a class of symmetric orthogonal polynomials in many variables. They include important classes of special functions such as Schur functions and Hall–Littlewood polynomials and play important roles in various fields of mathematics and mathematical physics. After an overview of Schur functions, the author introduces Macdonald polynomials (of type A, in the GLn version) as eigenfunctions of a q-difference operator, called the Macdonald–Ruijsenaars operator, in the ring of symmetric polynomials. Starting from this definition, various remarkable properties of Macdonald polynomials are explained, such as orthogonality, evaluation formulas, and self-duality, with emphasis on the roles of commuting q-difference operators. The author also explains how Macdonald polynomials are formulated in the framework of affine Hecke algebras and q-Dunkl operators.Table of ContentsOverview of Macdonald polynomials.- Preliminaries on symmetric functions.- Schur functions.- Macdonald polynomials: Definition and examples.- Orthogonality and higher order q-difference operators.- Self-duality, Pieri formula and Cauchy formulas.- Littlewood–Richardson coefficients and branching coefficients.- Affine Hecke algebra and q-Dunkl operators (overview).

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Book SynopsisAprende Álgebra I de forma clara, sencilla y efectiva. Álgebra I paso a paso para principiantes es la guía ideal para quienes desean dominar los conceptos fundamentales del álgebra sin estrés ni confusión.Escrito por el experto en educación Dr. Jacob Kohannim, este libro ofrece una explicación accesible de los temas más importantes de Álgebra I, presentados de manera progresiva y amigable. Dentro de este libro, encontrarás: Definiciones claras y ejemplos fáciles de entender Explicaciones paso a paso de ecuaciones, desigualdades, funciones y más Ejercicios prácticos diseñados para reforzar el aprendizaje Consejos y estrategias para resolver problemas de forma eficiente Actividades ideales para estudiantes, padres o autodidactas Ya sea que estés empezando a aprender álgebra, necesites reforzar tus bases, o estés apoyando a un estudiante, Álgebra I paso a paso para principiantes te acompañará en cada etapa del camino hacia el dominio de las matemáticas. Construye una base sólida en álgebra y prepárate para el éxito académico!

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Book SynopsisDomina Álgebra II con explicaciones claras y práctica guiada. Álgebra II Paso a Paso para...

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