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The cartoon guide to algebra
\"Covers all of algebra's essentials--including rational and real numbers, the number line, variables, expressions, laws of combination, linear and quadratic equations, rates, proportion, and graphing\"--Back cover.
Book
Weight Multiplicities and Young Tableaux Through Affine Crystals
The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is
hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine
Kac–Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of
classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the
eBook
Representation Theory of Geigle-Lenzing Complete Intersections
Weighted projective lines, introduced by Geigle and Lenzing in 1987, are important objects in representation theory. They have
tilting bundles, whose endomorphism algebras are the canonical algebras introduced by Ringel. The aim of this paper is to study their
higher dimensional analogs. First, we introduce a certain class of commutative Gorenstein rings
eBook
The mending of broken bones : a modern guide to classical algebra
\"The Mending of Broken Bones reveals the beauty of algebra, guiding even the mathematically disinclined toward the discipline's intellectual gratifications. Far from a mere classroom chore, algebra is a rich philosophical vein and a tool for the curious-a gateway to creative solutions, hidden patterns, and surprising unknowns.\"-- Provided by publisher.
BOOK
The Representation Theory of the Increasing Monoid
We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation
category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish
properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous
connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle
algebras.
eBook
EnVision algebra 1
\"EnVision A G A ©2018 is a brand-new high school mathematics program. It includes Algebra 1, Geometry, and Algebra 2. enVision A G A helps students look at math in new ways, with engaging, relevant, and adaptive content. For teachers, the program offers a flexible choice of options and resources. Customize instruction, practice, and assessments. Re-energize students and help them become more self-directed and independent learners\"--Provided by publisher
Book
Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements,
interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the
monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left
regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such
structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left
regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order
complexes of posets naturally associated to the left regular band.
The purpose of the present monograph is to further develop and
deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all
simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left
regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the
examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure
on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional
oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A
fairly complete picture of the representation theory for CW left regular bands is obtained.
eBook