Representation Theory of the Symmetric Group

2011 ◽  
pp. 117-129
Author(s):  
Benjamin Steinberg
Keyword(s):  
Representation Theory ◽  
Symmetric Group

scholarly journals A $\lambda$-ring Frobenius Characteristic for $G\wr S_n$

10.37236/1809 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Anthony Mendes ◽  
Jeffrey Remmel ◽  
Jennifer Wagner
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Kronecker Product ◽  
Reproducing Kernel ◽  
Irreducible Representations ◽  
Inner Product ◽  
Irreducible Components ◽  
Lambda Ring

A $\lambda$-ring version of a Frobenius characteristic for groups of the form $G \wr S_n$ is given. Our methods provide natural analogs of classic results in the representation theory of the symmetric group. Included is a method decompose the Kronecker product of two irreducible representations of $G\wr S_n$ into its irreducible components along with generalizations of the Murnaghan-Nakayama rule, the Hall inner product, and the reproducing kernel for $G\wr S_n$.

Induced Representations and Invariants

1950 ◽  
Vol 2 ◽  
pp. 334-343 ◽  
Author(s):  
G. DE B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Explicit Formula ◽  
Linear Group ◽  
Schur Functions ◽  
The Other ◽  
Induced Representations ◽  
Direct Sum ◽  
Invariant Matrices ◽  
The One

1. Introduction. The problem of the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices is intimately associated with the representation theory of the full linear group on the one hand and with the representation theory of the symmetric group on the other. In a previous paper the author gave an explicit formula for this reduction in terms of characters of the symmetric group. Later J. A. Todd derived the same formula using Schur functions, i.e. characters of representations of the full linear group.

On the Modular Representations of the Symmetric Group

1954 ◽  
Vol 6 ◽  
pp. 486-497 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Young Diagram ◽  
Modular Representation ◽  
Modular Representations ◽  
Modular Representation Theory

The study of the modular representation theory of the symmetric group has been greatly facilitated lately by the introduction of the graph (9, III ), the q-graph (5) and the hook-graph (4) of a Young diagram [λ]. In the present paper we seek to coordinate these ideas and relate them to the r-inducing and restricting processes (9, II ).

scholarly journals Modular Representations of Sn

1964 ◽  
Vol 16 ◽  
pp. 191-203 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Modular Representations ◽  
Modular Theory ◽  
Definition Of ◽  
Image Position

The purpose of this paper is to clarify and sharpen the argument in the last two chapters of the author's Representation theory of the symmetric group(3). When these chapters were written the peculiar properties of the case p = 2 were not fully appreciated. No difficulty arises in the definition of the block in terms of the p-core, or in the application of the general modular theory based on the formula

Interpretation of generalized matrix functions via geometric measure of quantum entanglement

2015 ◽  
Vol 13 (07) ◽  
pp. 1550049
Author(s):  
Haixia Chang ◽  
Vehbi E. Paksoy ◽  
Fuzhen Zhang
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Quantum Entanglement ◽  
Geometric Interpretation ◽  
Matrix Functions ◽  
Irreducible Characters ◽  
Geometric Measure ◽  
Generalized Matrix

By using representation theory and irreducible characters of the symmetric group, we introduce character dependent states and study their entanglement via geometric measure. We also present a geometric interpretation of generalized matrix functions via this entanglement analysis.

scholarly journals The Robinson―Schensted Correspondence and $A_2$-webs

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Matthew Housley ◽  
Heather M. Russell ◽  
Julianna Tymoczko
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Quantum Group ◽  
Classical Limit ◽  
Young Tableaux ◽  
Tensor Power ◽  
Recent Interest ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Graphical Algorithm

International audience The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensors powers of $V^+$, the standard representation of the quantum group. In particular, the invariant webs for the 3$n$th tensor power of $V^+$ correspond bijectively to $[n,n,n]$ standard Young tableaux. Kuperberg originally defined this map in terms of a graphical algorithm, and subsequent papers of Khovanov–Kuperberg (1999) and Tymoczko (2012) introduce algorithms for computing the inverse. The main result of this paper is a redefinition of Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan–Lusztig theory to show that Kuperberg's map is a direct analogue of the Robinson–Schensted correspondence.

A Remark by Philip Hall

1958 ◽  
Vol 1 (1) ◽  
pp. 21-23 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Linear Group ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Linear Transformations ◽  
Modern Physics ◽  
The Right ◽  
The Relationship

The relationship between the representation theory of the full linear group GL(d) of all non-singular linear transformations of degree d over a field of characteristic zero and that of the symmetric group Sn goes back to Schur and has been expounded by Weyl in his classical groups, [4; cf also 2 and 3]. More and more, the significance of continuous groups for modern physics is being pressed on the attention of mathematicians, and it seems worth recording a remark made to the author by Philip Hall in Edmonton.As is well known, the irreducible representations of Sn are obtainable from the Young diagrams [λ]=[λ1, λ2 ,..., λr] consisting of λ1 nodes in the first row, λ2 in the second row, etc., where λ1≥λ2≥ ... ≥λr and Σ λi = n. If we denote the jth node in the ith row of [λ] by (i,j) then those nodes to the right of and below (i,j), constitute, along with the (i,j) node itself, the (i,j)-hook of length hij.

scholarly journals Voting, the Symmetric Group, and Representation Theory

2009 ◽  
Vol 116 (8) ◽  
pp. 667-687 ◽  
Author(s):  
Zajj Daugherty ◽  
Alexander K. Eustis ◽  
Gregory Minton ◽  
Michael E. Orrison
Keyword(s):  
Representation Theory ◽  
Symmetric Group

scholarly journals Affine symmetric group

2021 ◽  
Vol 4 (1) ◽  
pp. 3
Author(s):  
Joel Brewster Lewis
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Mathematical Structure ◽  
Number Line ◽  
Geometric Description ◽  
Generators And Relations ◽  
Higher Dimensional ◽  
Finite Set ◽  
Finite Symmetric Group

The affine symmetric group is a mathematical structure that describes the symmetries of the number line and the regular triangular tesselation of the plane, as well as related higher dimensional objects. It is an infinite extension of the symmetric group, which consists of all permutations (rearrangements) of a finite set. In additition to its geometric description, the affine symmetric group may be defined as the collection of permutations of the integers (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. These different definitions allow for the extension of many important properties of the finite symmetric group to the infinite setting, and are studied as part of the fields of combinatorics and representation theory.

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