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Garnir Relations
In mathematics, the Garnir relations give a way of expressing a basis of the Specht modules ''V''λ in terms of standard polytabloids. Specht modules in terms of polytabloids Given a partition ''λ'' of ''n'', one has the Specht module ''V''λ. In characteristic 0, this is an irreducible representation of the symmetric group ''S''''n''. One can construct ''Vλ'' explicitly in terms of polytabloids as follows: * Start with the permutation representation of ''S''''n'' acting on all Young tableaux of shape ''λ'', which are fillings of the Young diagram of ''λ'' with numbers 1, 2, ... ''n'', each used once (note that we do not require the tableaux to be standard, there are no conditions imposed along rows or columns). The group ''S''''n'' acts by permuting the positions in each tableau (for instance there is a cyclic permutation the cycles the entries of the first row one place forward). * A Young tabloid is an orbit of Young tableaux under the action of the row permutations, the subg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Specht Module
In mathematics, a Specht module is one of the representations of symmetric groups studied by . They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of ''n'' form a complete set of irreducible representations of the symmetric group on ''n'' points. Definition Fix a partition λ of ''n'' and a commutative ring ''k''. The partition determines a Young diagram with ''n'' boxes. A Young tableau of shape λ is a way of labelling the boxes of this Young diagram by distinct numbers 1, \dots, n. A tabloid is an equivalence class of Young tableaux where two labellings are equivalent if one is obtained from the other by permuting the entries of each row. For each Young tableau ''T'' of shape λ let \ be the corresponding tabloid. The symmetric group on ''n'' points acts on the set of Young tableaux of shape λ. Consequently, it acts on tabloids, and on the free ''k''-module ''V'' with the tabloids as basis. Given a Young table ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition (number Theory)
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, can be partitioned in five distinct ways: : : : : : The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . A summand in a partition is also called a part. The number of partitions of is given by the partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general. Examples The seven partitions of 5 are: * 5 * 4 + 1 * 3 + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young Tableau
In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley. Definitions ''Note: this article uses the English convention for displaying Young diagrams and tableaux''. Diagrams A Young diagram (also called a Ferrers diagram, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young Tabloid
In mathematics, a Specht module is one of the representations of symmetric groups studied by . They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of ''n'' form a complete set of irreducible representations of the symmetric group on ''n'' points. Definition Fix a partition λ of ''n'' and a commutative ring ''k''. The partition determines a Young diagram with ''n'' boxes. A Young tableau of shape λ is a way of labelling the boxes of this Young diagram by distinct numbers 1, \dots, n. A tabloid is an equivalence class of Young tableaux where two labellings are equivalent if one is obtained from the other by permuting the entries of each row. For each Young tableau ''T'' of shape λ let \ be the corresponding tabloid. The symmetric group on ''n'' points acts on the set of Young tableaux of shape λ. Consequently, it acts on tabloids, and on the free ''k''-module ''V'' with the tabloids as basis. Given a Young tablea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Row Descent
Row or ROW may refer to: Exercise *Rowing, or a form of aquatic movement using oars *Row (weight-lifting), a form of weight-lifting exercise Math *Row vector, a 1 × ''n'' matrix in linear algebra. *Row (database), a single, implicitly structured data item in a table *Tone row, an arrangement of the twelve notes of the chromatic scale Other *Reality of Wrestling, an American professional wrestling promotion founded in 2005 * ''Row'' (album), an album by Gerard *Right-of-way (transportation), ROW, also often R/O/W. *The Row (fashion label) Places * Rów, Pomeranian Voivodeship, north Poland *Rów, Warmian-Masurian Voivodeship, north Poland *Rów, West Pomeranian Voivodeship, northwest Poland *Roswell International Air Center's IATA code * Row, a former spelling of Rhu, Dunbartonshire, Scotland *The Row (Lyme, New York), a set of historic homes *The Row, Virginia, an unincorporated community *Rest of the world or RoW See also *Row house *Controversy, sometimes called "row" in B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Row Descent And Garnir Element
Row or ROW may refer to: Exercise *Rowing, or a form of aquatic movement using oars *Row (weight-lifting), a form of weight-lifting exercise Math *Row vector, a 1 × ''n'' matrix in linear algebra. *Row (database), a single, implicitly structured data item in a table *Tone row, an arrangement of the twelve notes of the chromatic scale Other *Reality of Wrestling, an American professional wrestling promotion founded in 2005 * ''Row'' (album), an album by Gerard *Right-of-way (transportation), ROW, also often R/O/W. *The Row (fashion label) Places * Rów, Pomeranian Voivodeship, north Poland *Rów, Warmian-Masurian Voivodeship, north Poland *Rów, West Pomeranian Voivodeship, northwest Poland *Roswell International Air Center's IATA code * Row, a former spelling of Rhu, Dunbartonshire, Scotland *The Row (Lyme, New York), a set of historic homes *The Row, Virginia, an unincorporated community *Rest of the world or RoW See also *Row house *Controversy, sometimes called "row" in B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Straightening Of A Polytabloid
"Straightenin" is a song by American hip hop trio Migos. It was released through Quality Control and Motown on May 14, 2021, as the second single from their fourth studio album, ''Culture III''. Background In March 2021, Quavo previewed the song's music video on social media. The track and the accompanying video were released on May 14, 2021 with an announcement of the album ''Culture III''. Hong Kong rapper and producer Big Spoon accused Migos of copying his 2020 song "Magic Show 魔術表演", which he compared to "Straightenin" in an Instagram video, saying, "I composed and produced the song by myself, did all the rapping, and played all the instruments so there are no samples or loops taken from elsewhere. I tweaked their song to the same tempo and key for comparison." Composition The song features "triplet flows and ad libs" quickly rapped over "booming bass, crisp percussion and a woozy synth line". Quavo mentions acting alongside Robert De Niro in the upcoming film ''Wash ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Module
In algebra, a Weyl module is a representation of a reductive algebraic group, introduced by and named after Hermann Weyl. In characteristic 0 these representations are irreducible, but in positive characteristic they can be reducible, and their decomposition into irreducible components can be hard to determine. See also *Borel–Weil–Bott theorem *Garnir relations In mathematics, the Garnir relations give a way of expressing a basis of the Specht modules ''V''λ in terms of standard polytabloids. Specht modules in terms of polytabloids Given a partition ''λ'' of ''n'', one has the Specht module ''V''λ. In ... Further reading * * *{{eom, id=Weyl_module, first=R., last= Dipper Representation theory Algebraic groups ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruce Sagan
Bruce E. Sagan (born March 29, 1954, Chicago, Illinois, United States) is an American Professor of Mathematics at Michigan State University. He specializes in enumerative, algebraic, and topological combinatorics. He is also known as a musician, playing music from Scandinavia and the Balkans. Early life Bruce Eli Sagan is the son of Eugene Benjamin Sagan and Arlene Kaufmann Sagan. He grew up in Berkeley, California. He started playing classical violin at a young age under the influence of his mother who was a music teacher and conductor. He received his B.S. in mathematics (1974) from California State University, East Bay (then called California State University, Hayward). He received his Ph.D. in mathematics (1979) from the Massachusetts Institute of Technology. His doctoral thesis "Partially Ordered Sets with Hooklengths – an Algorithmic Approach" was supervised by Richard P. Stanley. He was Stanley's third doctoral student. During his graduate school years he also joined a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Alexander Green
James Alexander "Sandy" Green FRS (26 February 1926 – 7 April 2014) was a mathematician and Professor at the Mathematics Institute at the University of Warwick, who worked in the field of representation theory. Early life Sandy Green was born in February 1926 in Rochester, New York, but moved to Toronto with his emigrant Scottish parents later that year. The family returned to Britain in May 1935 when his father, Frederick C. Green, took up the Drapers Professorship of French at the University of Cambridge. Education Green was educated at the Perse School, Cambridge. He won a scholarship to the University of St Andrews and matriculated aged 16 in 1942. He took an ordinary BSc in 1944, and then, after scientific service in the war, was awarded a BSc Honours in 1947. He gained his PhD at St John's College, Cambridge in 1951, under the supervision of Philip Hall and David Rees. Career World War II In the summer of 1944, he was conscripted for national scientific service at t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
