Dual representation

In mathematics, if G is a group and ρ is a linear representation of it on the vector space V the dual representation ρ* is defined over the dual vector space V* as follows: ρ* is the transpose of ρ, that is, ρ* = ρT for all g ∈ G. The dual representation is known as the contragredient representation. If g is a Lie algebra and π is a representation of it on the vector space V the dual representation π* is defined over the dual vector space V* as follows: π* = −πT for all X ∈ g; the motivation for this definition is that Lie algebra representation associated to the dual of a Lie group representation is computed by the above formula. But the definition of the dual of a Lie algebra representation makes sense if it does not come from a Lie group representation. In both cases, the dual representation is a representation in the usual sense. If a representation is irreducible the dual representation is irreducible—but not isomorphic to the original representation. On the other hand, the dual of the dual of any representation is isomorphic to the original representation.

Let us consider a unitary representation ρ of a group G, let us work in an orthonormal basis. Thus, ρ maps G into the group of unitary matrices; the abstract transpose in the definition of the dual representation may be identified with the ordinary matrix transpose. Since the adjoint of a matrix is the complex conjugate of the transpose, the transpose is the conjugate of the adjoint. Thus, ρ ∗ is the complex conjugate of the adjoint of the inverse of ρ, but since ρ is assumed to be unitary, the adjoint of the inverse of ρ is just ρ. The upshot of this discussion is that when working with unitary representations in an orthonormal basis, ρ ∗ is just the complex conjugate of ρ. In the representation theory of SU, the dual of each irreducible representation does turn out to be isomorphic to the representation, but for the representations of SU, the dual of the irreducible representation with label is the irreducible representation with label. In particular, the standard three-dimensional representation of SU is not isomorphic to its dual.

In the theory of quarks in the physics literature, the standard representation and its dual are called " 3 " and " 3 ¯." More in the representation theory of semisimple Lie algebras, the weights of the dual representation are the negatives of the weights of the original representation. Now, for a given Lie algebra, if it should happen that operator − I is an element of the Weyl group the weights of every representation are automatically invariant under the map μ ↦ − μ. For such Lie algebras, every irreducible representation will be isomorphic to its dual. Lie algebras with this property include the odd orthogonal Lie algebras s o and the symplectic Lie algebras s p. If, for a given Lie algebra, − I is not in the Weyl group the dual of an irreducible representation will generically not be isomorphic to the original representation. To understand how this works, we note that there is always a unique Weyl group element w 0 mapping the negative of the fundamental Weyl chamber to the fundamental Weyl chamber.

If we have an irreducible representation with highest weight μ, the lowest weight of the dual representation will be − μ. It follows that the highest weight of the dual representation will be w 0 ⋅. Since we are assuming − I is not in the Weyl group, w 0 cannot be − I, which means that the map μ ↦ w 0 ⋅ {\di

Natural number

In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.

A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.

Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.

Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a

Root of unity

In mathematics, a root of unity called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, are important in number theory, the theory of group characters, the discrete Fourier transform. Roots of unity can be defined in any field. If the characteristic of the field is zero, they are complex numbers that are algebraic integers. In positive characteristic, they belong to a finite field, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains n nth roots of unity, except if n is a multiple of the characteristic of the field. An nth root of unity, where n is a positive integer, is a number z satisfying the equation z n = 1. Unless otherwise specified, the roots of unity may be taken to be complex numbers, in this case, the nth roots of unity are exp = cos 2 k π n + i sin 2 k π n, k = 0, 1, …, n − 1; however the defining equation of roots of unity is meaningful over any field F, this allows considering roots of unity in F.

Whichever is the field F, the roots of unity in F are either complex numbers, if the characteristic of F is 0, or, belong to a finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details. An nth root of unity is said to be primitive if it is not a kth root of unity for some smaller k, if z n = 1 and z k ≠ 1 for k = 1, 2, 3, …, n − 1. If n is a prime number, all nth roots of unity, except 1, are primitive. In the above formula in terms of exponential and trigonometric functions, the primitive nth roots of unity are those for which k and n are coprime integers. Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see Finite field § Roots of unity. For the case of roots of unity in rings of modular integers, see Root of unity modulo n; every nth root of unity z is a primitive ath root of unity for some a ≤ n, the smallest positive integer such that za = 1.

Any integer power of an nth root of unity is an nth root of unity, as n = z k n = k = 1 k = 1. This is true for negative exponents. In particular, the reciprocal of an nth root of unity is its complex conjugate, is an nth root of unity: 1 z = z − 1 = z n − 1 = z ¯. If z is an nth root of unity and a ≡ b za = zb. In fact, by the definition of congruence, a = b + kn for some integer k, z a = z b + k n = z b z k n = z b k = z b 1 k = z b. Therefore, given a power za of z, one has za = zr, where 0 ≤ r < n is the remainder of the Euclidean division of a by n. Let z be a primitive nth root of unity; the powers z, z2, ... zn−1, zn = z0 = 1 are nth root of unity and are all distinct. This implies that z, z2, ... zn−1, zn = z0 = 1 are all of the nth roots of unity, since an nth-degree polynomial equation has at most n distinct solutions. From the preceding, it follows that, if z is a primitive nth root of unity z a = z b if and only if a ≡ b. If z is not primitive a ≡ b implies z a = z b, but the converse may be false, as shown by the following example.

If n = 4, a non-primitive nth root of unity is z = –1, one has z 2 =

Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques from commutative algebra, for solving geometrical problems about these sets of zeros; the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, parabolas, hyperbolas, cubic curves like elliptic curves, quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.

Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis and number theory. A study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, it becomes more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution. In the 20th century, algebraic geometry split into several subareas; the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more to the points with coordinates in an algebraically closed field. Real algebraic geometry is the study of the real points of an algebraic variety. Diophantine geometry and, more arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, p-adic fields.

A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers, it consists of algorithm design and software development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way, similar to its use in the study of differential and analytic manifolds; this is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring.

This means that a point of such a scheme may be either a subvariety. This approach enables a unification of the language and the tools of classical algebraic geometry concerned with complex points, of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach. In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 − 1 = 0. A "slanted" circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 − 1 = 0, x + y + z = 0. First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed.

We consider the affine space of dimension n over denoted An. When one fixes a coordinate system, one may identify An with kn; the purpose of not working with kn is to emphasize that one "forgets" the vector space structure that kn carries. A function f: An → A1 is said to be polynomial if it can be written as a polynomial, that is, if there is a polynomial p in k such that f = p for every point M with coordinates in An; the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the

John Tate

John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He is professor emeritus at Harvard University, he was awarded the Abel Prize in 2010. Tate was born in Minneapolis, his father, John Tate Sr. was a professor of physics at the University of Minnesota, a longtime editor of Physical Review. His mother, Lois Beatrice Fossler, was a high school English teacher. Tate Jr. received his bachelor's degree in mathematics from Harvard University, entered the doctoral program in physics at Princeton University. He transferred to the mathematics department and received his PhD in 1950 as a student of Emil Artin. Tate taught at Harvard for 36 years before joining the University of Texas in 1990 as a Sid W. Richardson Foundation Regents Chair, he retired from the Texas mathematics department in 2009, returned to Harvard as a professor emeritus. He resides in Cambridge, Massachusetts with his wife Carol.

He has three daughters with his first wife Karin Tate. Tate's thesis on Fourier analysis in number fields has become one of the ingredients for the modern theory of automorphic forms and their L-functions, notably by its use of the adele ring, its self-duality and harmonic analysis on it. Together with his teacher Emil Artin, Tate gave a cohomological treatment of global class field theory, using techniques of group cohomology applied to the idele class group and Galois cohomology; this treatment made more transparent some of the algebraic structures in the previous approaches to class field theory which used central division algebras to compute the Brauer group of a global field. Subsequently, Tate introduced. In the decades following that discovery he extended the reach of Galois cohomology with the Poitou–Tate duality, the Tate–Shafarevich group, relations with algebraic K-theory. With Jonathan Lubin, he recast local class field theory by the use of formal groups, creating the Lubin–Tate local theory of complex multiplication.

He has made a number of individual and important contributions to p-adic theory. He found a p-adic analogue of Hodge theory, now called Hodge–Tate theory, which has blossomed into another central technique of modern algebraic number theory. Other innovations of his include the'Tate curve' parametrization for certain p-adic elliptic curves and the p-divisible groups. Many of his results were not published and some of them were written up by Serge Lang, Jean-Pierre Serre, Joseph H. Silverman and others. Tate and Serre collaborated on a paper on good reduction of abelian varieties; the classification of abelian varieties over finite fields was carried out by Taira Tate. The Tate conjectures are the equivalent for étale cohomology of the Hodge conjecture, they relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of'Tate cycles' that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the general case, was involved in the proof of the Mordell conjecture by Gerd Faltings.

Tate has had a major influence on the development of number theory through his role as a Ph. D. advisor. His students include George Bergman, Bernard Dwork, Benedict Gross, Robert Kottwitz, Jonathan Lubin, Stephen Lichtenbaum, James Milne, V. Kumar Murty, Carl Pomerance, Ken Ribet, Joseph H. Silverman, Dinesh Thakur. In 1956 Tate was awarded the American Mathematical Society's Cole Prize for outstanding contributions to number theory. In 1995 he received the Leroy P. Steele Prize for Lifetime Achievement from the American Mathematical Society, he was awarded a Wolf Prize in Mathematics in 2002/03 for his creation of fundamental concepts in algebraic number theory. In 2012 he became a fellow of the American Mathematical Society. In 2010, the Norwegian Academy of Science and Letters, of which he is a member, awarded him the Abel Prize, citing "his vast and lasting impact on the theory of numbers". According to a release by the Abel Prize committee "Many of the major lines of research in algebraic number theory and arithmetic geometry are only possible because of the incisive contributions and illuminating insights of John Tate.

He has left a conspicuous imprint on modern mathematics."Tate has been described as "one of the seminal mathematicians for the past half-century" by William Beckner, Chairman of the Department of Mathematics at the University of Texas. Tate, Fourier analysis in number fields and Hecke's zeta functions, Princeton University Ph. D. thesis under Emil Artin. Reprinted in Cassels, J. W. S.. Algebraic number theory, London: Academic Press, pp. 305–347, MR 0215665 Tate, John, "The higher dimensional cohomology groups of class field theory", Ann. of Math. 2, 56: 294–297, doi:10.2307/1969801, MR 0049950 Lang, Serge.

Number theory

Number theory is a branch of pure mathematics devoted to the study of the integers. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers. Integers can be considered either as solutions to equations. Questions in number theory are best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may study real numbers in relation to rational numbers, for example, as approximated by the latter; the older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory"; the use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.

The first historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 contains a list of "Pythagorean triples", that is, integers such that a 2 + b 2 = c 2. The triples are too large to have been obtained by brute force; the heading over the first column reads: "The takiltum of the diagonal, subtracted such that the width..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity 2 + 1 = 2, implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and reordered by c / a for actual use as a "table", for example, with a view to applications, it is not known whether there could have been any. It has been suggested instead. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra was exceptionally well developed. Late Neoplatonic sources state.

Much earlier sources state that Pythagoras traveled and studied in Egypt. Euclid IX 21–34 is probably Pythagorean. Pythagorean mystics gave great importance to the even; the discovery that 2 is irrational is credited to the early Pythagoreans. By revealing that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; this forced a distinction between numbers, on the one hand, lengths and proportions, on the other hand. The Pythagorean tradition spoke of so-called polygonal or figurate numbers. While square numbers, cubic numbers, etc. are seen now as more natural than triangular numbers, pentagonal numbers, etc. the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period. We know of no arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both; the Chinese remainder theorem appears as an exercise in Sunzi Suanjing There is some numerical mysticism in Chinese mathematics, unlike that of the Pythagoreans, it seems to have led nowhere.

Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation. Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-m