Some combinatorial results involving Young diagrams

1978 ◽  
Vol 83 (1) ◽  
pp. 1-10 ◽  
Author(s):  
G. D. James
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Symmetric Group ◽  
Young Diagram ◽  
New Method ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Sufficient Criterion ◽  
Power Diagram ◽  
Necessary And Sufficient

In the first half of this paper we introduce a new method of examining the q-hook structure of a Young diagram, and use it to prove most of the standard results about q-cores and q-quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given q-weight and says how many of them are q-regular. In the case where q is prime, this tells us how many ordinary and q-modular irreducible representations of the symmetric group there are in a given q-block. None of the results of section 2 is original. In the next section we give a new definition, the p-power diagram, which is closely connected with the p-quotient. This concept is interesting because when p is prime a condition involving the p-power diagram appears to be a necessary and sufficient criterion for the diagram to be p-regular and the corresponding ordinary irreducible representation of to remain irreducible modulo p. In this paper we derive combinatorial results involving the p-power diagram, and in a later article we investigate the relevant representation theory.

scholarly journals Search for Young diagrams with large dimensions

2019 ◽  
pp. 33-43
Author(s):  
Vasilii S. Duzhin ◽  
◽  
Anastasia A. Chudnovskaya ◽  
Keyword(s):  
Symmetric Group ◽  
Young Diagram ◽  
Numerical Experiments ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Maximum Dimension ◽  
Asymptotic Combinatorics

Search for Young diagrams with maximum dimensions or, equivalently, search for irreducible representations of the symmetric group $S(n)$ with maximum dimensions is an important problem of asymptotic combinatorics. In this paper, we propose algorithms that transform a Young diagram into another one of the same size but with a larger dimension. As a result of massive numerical experiments, the sequence of $10^6$ Young diagrams with large dimensions was constructed. Furthermore, the proposed algorithms do not change the first 1000 elements of this sequence. This may indicate that most of them have the maximum dimension. It has been found that the dimensions of all Young diagrams of the resulting sequence starting from the 75778th exceed the dimensions of corresponding diagrams of the greedy Plancherel sequence.

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.

Star Diagrams and the Symmetric Group

1950 ◽  
Vol 2 ◽  
pp. 79-92 ◽  
Author(s):  
R. A. Staal
Keyword(s):  
Symmetric Group ◽  
Young Diagram ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
One To One

Introduction. The irreducible representations of the symmetric group Sn, were shown by A. Young to be in one-to-one correspondence with certain arrays of n nodes. E.g. for n = 12 and the partition λ = [4, 4, 3, 1] we have the array which we call a “Young diagram.” The question arises as to the manner in which various properties of the representations are reflected in their corresponding Young diagrams.

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 $G\wr S_n$.

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 Extensions of differential representations of SL2 and tori

2012 ◽  
Vol 12 (1) ◽  
pp. 199-224 ◽  
Author(s):  
Andrey Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Difference Equations ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Irreducible Representations ◽  
Direct Sums ◽  
Rational Representation ◽  
Linear Differential ◽  
Linear Differential Algebraic Groups

AbstractLinear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with ${\mathbf{SL} }_{2} $ and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of ${\mathbf{SL} }_{2} $. In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.

scholarly journals On the Kronecker Product $s_{(n-p,p)}\ast s_{\lambda}$

10.37236/1925 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
C. M. Ballantine ◽  
R. C. Orellana
Keyword(s):  
Irreducible Representation ◽  
Tensor Product ◽  
Symmetric Group ◽  
Kronecker Product ◽  
Schur Functions ◽  
Irreducible Representations ◽  
Kronecker Coefficient

The Kronecker product of two Schur functions $s_{\lambda}$ and $s_{\mu}$, denoted $s_{\lambda}\ast s_{\mu}$, is defined as the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group indexed by partitions of $n$, $\lambda$ and $\mu$, respectively. The coefficient, $g_{\lambda,\mu,\nu}$, of $s_{\nu}$ in $s_{\lambda}\ast s_{\mu}$ is equal to the multiplicity of the irreducible representation indexed by $\nu$ in the tensor product. In this paper we give an algorithm for expanding the Kronecker product $s_{(n-p,p)}\ast s_{\lambda}$ if $\lambda_1-\lambda_2\geq 2p$. As a consequence of this algorithm we obtain a formula for $g_{(n-p,p), \lambda ,\nu}$ in terms of the Littlewood-Richardson coefficients which does not involve cancellations. Another consequence of our algorithm is that if $\lambda_1-\lambda_2\geq 2p$ then every Kronecker coefficient in $s_{(n-p,p)}\ast s_{\lambda}$ is independent of $n$, in other words, $g_{(n-p,p),\lambda,\nu}$ is stable for all $\nu$.

On a conjecture of Carter concerning irreducible Specht modules

1978 ◽  
Vol 83 (1) ◽  
pp. 11-17 ◽  
Author(s):  
G. D. James
Keyword(s):  
Irreducible Representation ◽  
Symmetric Group ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Specht Module ◽  
Specht Modules ◽  
Definition Of

We study the question: Which ordinary irreducible representations of the symmetric group remain irreducible modulo a prime p?Let Sλ be the Specht module corresponding to the partition λ of n. The definition of Sλ is ‘independent of the field we are working over’. When the field has characteristic zero, Sλ is irreducible, and gives the ordinary irreducible representation of corresponding to the partition λ. Thus we are interested in the problem of whether or not Sλ is irreducible over a field of characteristic p.

On the Irreducible Representations of the Symmetric Group

1952 ◽  
Vol 4 ◽  
pp. 381-384
Author(s):  
Masaru Osima
Keyword(s):  
Symmetric Group ◽  
Young Diagram ◽  
Irreducible Representations ◽  
Image Position

Let T be a Young diagram of n nodes:(1)ai being the length of its ith row. With respect to a prime p, we denote by T0 the p-core of T. If T0 consists of m nodes, then(2) m=n-lp,where l is the number of successive p-hooks [3] removable from T to yield its p-core T0. We have stated in [4] the following theorem :

An Example on Computing the Irreducible Representation of Finite Metacyclic Groups by Using Great Orthogonality Theorem Method

2013 ◽  
Vol 64 (1) ◽  
Author(s):  
Nizar Majeed Samin ◽  
Nor Haniza Sarmin ◽  
Hamisan Rahmat
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Abstract Algebra ◽  
Irreducible Representations ◽  
Metacyclic Groups ◽  
Orthogonality Theorem ◽  
Axiomatic Systems

Representation theory is a study of real realizations of the axiomatic systems of abstract algebra. For any group, the number of possible representative sets of matrices is infinite, but they can all be reduced to a single fundamental set, called the irreducible representations of the group. This paper focuses on an example of finite metacyclic groups of class two of order 16. The irreducible representation of that group is found by using Great Orthogonality Theorem Method.

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