Symmetric group

About: Symmetric group is a research topic. Over the lifetime, 7945 publications have been published within this topic receiving 134099 citations.

Coxeter graphs and towers of algebras

Abstract: 1. Matrices over the natural numbers: values of the norm, classification, and variations.- 1.1. Introduction.- 1.2. Proof of Kronecker's theorem.- 1.3. Decomposability and pseudo-equivalence.- 1.4. Graphs with norms no larger than 2.- 1.5. The set E of norms of graphs and integral matrices.- 2. Towers of multi-matrix algebras.- 2.1. Introduction.- 2.2. Commutant and bicommutant.- 2.3. Inclusion matrix and Bratteli diagram for inclusions of multi-matrix algebras.- 2.4. The fundamental construction and towers for multi-matrix algebras.- 2.5. Traces.- 2.6. Conditional expectations.- 2.7. Markov traces on pairs of multi-matrix algebras.- 2.8. The algebras A?,k for generic ?.- 2.9. An approach to the non-generic case.- 2.10. A digression on Hecke algebras.- 2.10.a. The complex Hecke algebra defined by GLn(q) and its Borel subgroup.- 2.10.b. The Hecke algebras Hq,n.- 2.10.c. Complex representations of the symmetric group.- 2.10.d. Irreducible representations of Hq,n for q ? ?.- 2.11. The relationship between A?,n and the Hecke algebras.- 3. Finite von Neumann algebras with finite dimensional centers.- 3.1. Introduction.- 3.2. The coupling constant: definition.- 3.3. The coupling constant: examples.- 3.3.a. Discrete series.- 3.3.b. Factors defined by icc groups.- 3.3.c. W*(?)-modules associated to subrepresentations of ?G.- 3.3.d. The formula dim?(H) = covol(?) d?.- 3.3.e. A digression on the Peterson inner product.- 3.4. Index for subfactors of II1 factors.- 3.5. Inclusions of finite von Neumann algebras with finite dimensional centers.- 3.6. The fundamental construction.- 3.7. Markov traces on EndN(M), a generalization of index.- 4. Commuting squares, subfactors, and the derived tower.- 4.1. Introduction.- 4.2. Commuting squares.- 4.3. Wenzl's index formula.- 4.4. Examples of irreducible pairs of factors of index less than 4, and a lemma of C. Skau.- 4.5. More examples of irreducible paris of factors, and the index value 3 + 31/2.- 4.6. The derived tower and the Coxeter invariant.- 4.7. Examples of derived towers.- 4.7.a. Finite group actions.- 4.7.b. The An Coxeter graphs.- 4.7.c. A general method.- 4.7.d. Some examples of derived towers for index 4 subfactors.- 4.7.e. The tunnel construction.- 4.7.f. The derived tower for R ? R?, when ? > 4.- Appendix I. Classification of Coxeter graphs with spectral radius just beyond the Kronecker range.- I.1. The results.- I.2. Computations of characteristic polynomials for ordinary graphs.- I.3. Proofs of theorems I.1.2 and I.1.3.- Appendix II.a. Complex semisimple algebras and finite dimensional C*-algebras.- Appendix III. Hecke groups and other subgroups of PSL(2,?) generated by parabolic pairs.- References.

The representation theory of finite groups

Abstract: Group Algebras. The Green and Brauer Correspondences, the First Main Theorem. Characters, Higher Decomposition Numbers, the Second Main Theorem. Subsections, Lower Defect Groups, the Third Main Theorem. Blocks and Extensions of R. Blocks with a Cyclic Defect Group. Groups with a Sylow Group of Prime Order. The Structure of A(G). p-Solvable Groups. An Analogue of Jordan's Theorem. Types of Blocks.

The Theory of Group Characters and Matrix Representations of Groups

Abstract: Matrices Algebras Groups The Frobenius algebra The symmetric group Immanants and $S$-functions $S$-functions of special series The calculation of the characters of the symmetric group Group characters and the structure of groups Continuous matrix groups and invariant matrices Groups of unitary matrices Appendix Bibliography Supplementary bibliography Index.

Braids, link polynomials and a new algebra

Abstract: A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on 2 parameters. The decomposition of the corresponding algebras into irreducible components is given and it is shown how they are related to Jones' algebras and to Brauer's centralizer algebras. In [J,3] Vaughan Jones announced the discovery of a new polynomial invariant of knots and links, which bore many similarities to the classical Alexander polynomial, but was seen to detect properties of a link which could not be detected by the Alexander invariants. The discovery was a real surprise, one of those exciting moments in mathematics when two seemingly unrelated disciplines turn out to have deep interconnections. The discovery came about in the following way. Jones' earlier contributions in the area of Operator Algebras had produced, in [J,1], a family of algebras An(t), t E C, indexed by the natural numbers n = 1 , 2, 3, . .. , and equipped with a trace function T: An(t) -C. His algebra An(t) was a quotient of the well-known Hecke algebra of the symmetric group, which we denote by 2 (1, m) to delineate our particular 2-parameter version of it. Jones had discovered, in [J,2], that there were representations of Artin's braid group Bn in the algebra An(t), in fact there were maps Bn 0 Atn(m A(t) from Bn into the multiplicative group of An (t) which factored through Links enter the picture via braids. Each oriented link L in oriented S3 can be represented by a (nonunique) element fl in some braid group Bn. There is an equivalence relation on B = H0= Bn known as Markov equivalence, which determines a 1-1 correspondence between equivalence classes [fl] E B and isotopy types of the associated oriented links Lfl. Jones' discovery was that with a small renormalization his trace function on A o(t) = Hln=l An(t) could be made into a function which lifted to an invariant on Markov classes Received by the editors December 9, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25; Secondary 20F29, 20C07. The work of the first author was supported in part by NSF grant #DMS-8503758. The work of the second author was supported in part by NSF grant #DMS-8510816. ? 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page

The Hook Graphs of the Symmetric Group

Abstract: Each irreducible representation [λ] of the symmetric group S n may be identified by a partition [λ] of n into non-negative integral parts λ1 ≥ λ2 ≥ … λ n ≥ 0, of which the first λ'j parts are ≥j, or by a right (Young) diagram also called [λ], that contains λi nodes in its ith row and λ'j nodes in its jth column.