Addendum to "On the distribution of the length of the second row of a Young diagram under Plancherel measure"

2000 ◽  
Vol 10 (6) ◽  
pp. 1606-1607 ◽  
Author(s):  
J. Baik ◽  
P. Deift ◽  
K. Johansson
Keyword(s):  
Young Diagram ◽  
Plancherel Measure

scholarly journals Permutations Without Long Decreasing Subsequences and Random Matrices

10.37236/929 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Piotr Šniady
Keyword(s):  
Random Matrices ◽  
Young Diagram ◽  
Random Matrix ◽  
Random Permutation ◽  
Fixed Number ◽  
Plancherel Measure ◽  
Young Diagrams ◽  
Gaussian Unitary Ensemble ◽  
Longest Increasing Subsequence ◽  
Unitary Ensemble

We study the shape of the Young diagram $\lambda$ associated via the Robinson–Schensted–Knuth algorithm to a random permutation in $S_n$ such that the length of the longest decreasing subsequence is not bigger than a fixed number $d$; in other words we study the restriction of the Plancherel measure to Young diagrams with at most $d$ rows. We prove that in the limit $n\to\infty$ the rows of $\lambda$ behave like the eigenvalues of a certain random matrix (namely the traceless Gaussian Unitary Ensemble random matrix) with $d$ rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix.

scholarly journals NILPOTENT ORBITS AND CODIMENSION-2 DEFECTS OF 6d${\mathcal N} = (2, 0)$ THEORIES

2013 ◽  
Vol 28 (03n04) ◽  
pp. 1340006 ◽  
Author(s):  
OSCAR CHACALTANA ◽  
JACQUES DISTLER ◽  
YUJI TACHIKAWA
Keyword(s):  
Lie Algebra ◽  
Young Diagram ◽  
Central Charge ◽  
Outer Automorphism ◽  
Natural Generalization ◽  
Nilpotent Orbits ◽  
Coulomb Branch ◽  
Local Properties

We study the local properties of a class of codimension-2 defects of the 6d [Formula: see text] theories of type J = A, D, E labeled by nilpotent orbits of a Lie algebra [Formula: see text], where [Formula: see text] is determined by J and the outer-automorphism twist around the defect. This class is a natural generalization of the defects of the six-dimensional (6d) theory of type SU (N) labeled by a Young diagram with N boxes. For any of these defects, we determine its contribution to the dimension of the Higgs branch, to the Coulomb branch operators and their scaling dimensions, to the four-dimensional (4d) central charges a and c and to the flavor central charge k.

Unimodularity and atomic Plancherel measure

1984 ◽  
Vol 266 (4) ◽  
pp. 513-518 ◽  
Author(s):  
L. Baggett
Keyword(s):  
Plancherel Measure

scholarly journals On the ${\cal S}_{n}$-Modules Generated by Partitions of a Given Shape

10.37236/835 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Daniel Kane ◽  
Steven Sivek
Keyword(s):  
Young Diagram ◽  
Module Homomorphism

Given a Young diagram $\lambda$ and the set $H^{\lambda}$ of partitions of $\{1,2,\dots$, $|\lambda|\}$ of shape $\lambda$, we analyze a particular ${\cal S}_{|\lambda|}$-module homomorphism ${\Bbb Q}H^{\lambda}\to{\Bbb Q}H^{\lambda'}$ to show that ${\Bbb Q}H^{\lambda}$ is a submodule of ${\Bbb Q}H^{\lambda'}$ whenever $\lambda$ is a hook $(n,1,1,\dots,1)$ with $m$ rows, $n\geq m$, or any diagram with two rows.

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.

scholarly journals Hook lengths in a skew Young diagram

10.37236/1309 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Young Diagram ◽  
Character Degrees ◽  
Young Diagrams ◽  
Inductive Proof

Regev and Vershik (Electronic J. Combinatorics 4 (1997), #R22) have obtained some properties of the set of hook lengths for certain skew Young diagrams, using asymptotic calculations of character degrees. They also conjectured a stronger form of one of their results. We give a simple inductive proof of this conjecture. Very recently, Regev and Zeilberger (Annals of Combinatorics, to appear) have independently proved this conjecture.

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