|
Dudley E. Littlewood
Dudley Ernest Littlewood (7 September 1903, London – 6 October 1979, Llandudno) was a British mathematician known for his work in group representation theory. He read mathematics at Trinity College, Cambridge, where his tutor was John Edensor Littlewood (they were not related). He was a lecturer at University College, Swansea from 1928 to 1947, and in 1948 took up the chair of mathematics at University College of North Wales, Bangor, retiring in 1970. He worked on invariant theory and group representation theory, especially of the symmetric group, often in collaboration with Archibald Read Richardson of Swansea. They introduced the immanant of a matrix, studied Schur functions and developed the Littlewood–Richardson rule for their multiplication. Littlewood was also interested in the application of representation theory to quantum mechanics. Selected publications *''The theory of group characters and matrix representations of groups.'' 1940; 2nd edition Oxford 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dudley E
Dudley is a large market town and administrative centre in the county of West Midlands, England, southeast of Wolverhampton and northwest of Birmingham. Historically an exclave of Worcestershire, the town is the administrative centre of the Metropolitan Borough of Dudley; in 2011 it had a population of 79,379. The Metropolitan Borough, which includes the towns of Stourbridge and Halesowen, had a population of 312,900. In 2014 the borough council named Dudley as the capital of the Black Country. Originally a market town, Dudley was one of the birthplaces of the Industrial Revolution and grew into an industrial centre in the 19th century with its iron, coal, and limestone industries before their decline and the relocation of its commercial centre to the nearby Merry Hill Shopping Centre in the 1980s. Tourist attractions include Dudley Zoo and Castle, the 12th century priory ruins, and the Black Country Living Museum. History Early history Dudley has a history dating bac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials. Definition (Jacobi's bialternant formula) Schur polynomials are indexed by integer partitions. Given a partition , where , and each is a non-negative integer, the functions a_ (x_1, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1976 Deaths
Events January * January 3 – The International Covenant on Economic, Social and Cultural Rights enters into force. * January 5 – The Pol Pot regime proclaims a new constitution for Democratic Kampuchea. * January 11 – The 1976 Philadelphia Flyers–Red Army game results in a 4–1 victory for the National Hockey League's Philadelphia Flyers over HC CSKA Moscow of the Soviet Union. * January 16 – The trial against jailed members of the Red Army Faction (the West German extreme-left militant Baader–Meinhof Group) begins in Stuttgart. * January 18 ** Full diplomatic relations are established between Bangladesh and Pakistan 5 years after the Bangladesh Liberation War. ** The Scottish Labour Party is formed as a breakaway from the UK-wide party. ** Super Bowl X in American football: The Pittsburgh Steelers defeat the Dallas Cowboys, 21–17, in Miami. * January 21 – First commercial Concorde flight, from London to Bahrain. * January 27 ** The United States v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1903 Births
Nineteen or 19 may refer to: * 19 (number), the natural number following 18 and preceding 20 * one of the years 19 BC, AD 19, 1919, 2019 Films * ''19'' (film), a 2001 Japanese film * ''Nineteen'' (film), a 1987 science fiction film Music * 19 (band), a Japanese pop music duo Albums * ''19'' (Adele album), 2008 * ''19'', a 2003 album by Alsou * ''19'', a 2006 album by Evan Yo * ''19'', a 2018 album by MHD * ''19'', one half of the double album ''63/19'' by Kool A.D. * ''Number Nineteen'', a 1971 album by American jazz pianist Mal Waldron * ''XIX'' (EP), a 2019 EP by 1the9 Songs * "19" (song), a 1985 song by British musician Paul Hardcastle. * "Nineteen", a song by Bad4Good from the 1992 album '' Refugee'' * "Nineteen", a song by Karma to Burn from the 2001 album ''Almost Heathen''. * "Nineteen" (song), a 2007 song by American singer Billy Ray Cyrus. * "Nineteen", a song by Tegan and Sara from the 2007 album '' The Con''. * "XIX" (song), a 2014 song by Slipknot. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Restricted Representation
In group theory, restriction forms a representation of a subgroup using a known representation of the whole group. Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to understand. Rules for decomposing the restriction of an irreducible representation into irreducible representations of the subgroup are called branching rules, and have important applications in physics. For example, in case of explicit symmetry breaking, the symmetry group of the problem is reduced from the whole group to one of its subgroups. In quantum mechanics, this reduction in symmetry appears as a splitting of degenerate energy levels into multiplets, as in the Stark or Zeeman effect. The induced representation is a related operation that forms a representation of the whole group from a representation of a subgroup. The relation between restriction and induction is described by Frobenius reciprocity and the Mackey theorem. Restrict ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plethysm
In algebra, plethysm is an operation on symmetric functions introduced by Dudley E. Littlewood, who denoted it by ⊗ . The word "plethysm" for this operation (after the Greek word πληθυσμός meaning "multiplication") was introduced later by , who said that the name was suggested by M. L. Clark. If symmetric functions are identified with operations in lambda rings, then plethysm corresponds to composition of operations. In representation theory Let ''V'' be a vector space over the complex numbers, considered as a representation of the general linear group GL(''V''). Each Young diagram λ corresponds to a Schur functor ''L''λ(-) on the category of GL(''V'')-representations. Given two Young diagrams λ and μ, consider the decomposition of ''L''λ(Lμ(''V'')) into a direct sum of irreducible representations of the group. By the representation theory of the general linear group we know that each summand is isomorphic to L_\nu(V) for a Young diagram \nu. So for so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall–Littlewood Polynomials
In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter ''t'' and a partition λ. They are Schur functions when ''t'' is 0 and monomial symmetric functions when ''t'' is 1 and are special cases of Macdonald polynomials. They were first defined indirectly by Philip Hall using the Hall algebra, and later defined directly by Dudley E. Littlewood (1961). Definition The Hall–Littlewood polynomial ''P'' is defined by :P_\lambda(x_1,\ldots,x_n;t) = \left( \prod_ \prod_^ \frac \right) , where λ is a partition of at most ''n'' with elements λ''i'', and ''m''(''i'') elements equal to ''i'', and ''S''''n'' is the symmetric group of order ''n''!. As an example, : P_(x_1,x_2;t) = x_1^4 x_2^2 + x_1^2 x_2^4 + (1-t) x_1^3 x_2^3 Specializations We have that P_\lambda(x;1) = m_\lambda(x), P_\lambda(x;0) = s_\lambda(x) and P_\lambda(x;-1) = P_\lambda(x) where the latter is the Schur ''P'' polynomials. Properties Expanding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Littlewood–Richardson Rule
In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of finite-dimensional representations of general linear groups, or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials. Littlewood–Richardson coefficients depend on three partitions, say \lambda,\mu,\nu, of which \lambda and \mu describe the Schur functions being multiplied, and \nu gives the Schur function of which this is the coefficient in the linear combination; in other words they ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Immanant Of A Matrix
In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent. Let \lambda=(\lambda_1,\lambda_2,\ldots) be a partition of an integer n and let \chi_\lambda be the corresponding irreducible representation-theoretic character of the symmetric group S_n. The ''immanant'' of an n\times n matrix A=(a_) associated with the character \chi_\lambda is defined as the expression :\operatorname_\lambda(A)=\sum_ \chi_\lambda(\sigma) a_ a_ \cdots a_. Examples The determinant is a special case of the immanant, where \chi_\lambda is the alternating character \sgn, of ''S''''n'', defined by the parity of a permutation. The permanent is the case where \chi_\lambda is the trivial character, which is identically equal to 1. For example, for 3 \times 3 matrices, there are three irreducible representations of S_3, as shown in the character table: As stated above, \chi_1 produces th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Llandudno
Llandudno (, ) is a seaside resort, town and community in Conwy County Borough, Wales, located on the Creuddyn peninsula, which protrudes into the Irish Sea. In the 2011 UK census, the community – which includes Gogarth, Penrhyn Bay, Craigside, Glanwydden, Penrhynside, and Bryn Pydew – had a population of 20,701. The town's name means "Church of Saint Tudno". Llandudno is the largest seaside resort in Wales, and as early as 1861 was being called 'the Queen of the Welsh Watering Places' (a phrase later also used in connection with Tenby and Aberystwyth; the word 'resort' came a little later). Historically a part of Caernarfonshire, Llandudno was formerly in the district of Aberconwy within Gwynedd. History The town of Llandudno developed from Stone Age, Bronze Age and Iron Age settlements over many hundreds of years on the slopes of the limestone headland, known to seafarers as the Great Orme and to landsmen as the Creuddyn Peninsula. The origins in recorded history are wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
