scholarly journals Algorithmically Distinguishing Irreducible Characters of the Symmetric Group

10.37236/9753 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Timothy Y. Chow ◽  
Jennifer Paulhus
Keyword(s):  
Symmetric Group ◽  
Irreducible Character ◽  
Irreducible Characters ◽  
Oracle Access

Suppose that $\chi_\lambda$ and $\chi_\mu$ are distinct irreducible characters of the symmetric group $S_n$. We give an algorithm that, in time polynomial in $n$, constructs $\pi\in S_n$ such that $\chi_\lambda(\pi)$ is provably different from $\chi_\mu(\pi)$. In fact, we show a little more. Suppose $f = \chi_\lambda$ for some irreducible character $\chi_\lambda$ of $S_n$, but we do not know $\lambda$, and we are given only oracle access to $f$. We give an algorithm that determines $\lambda$, using a number of queries to $f$ that is polynomial in $n$. Each query can be computed in time polynomial in $n$ by someone who knows $\lambda$.

scholarly journals On Multiple Transitivity of Permutation Groups

1961 ◽  
Vol 18 ◽  
pp. 93-109 ◽  
Author(s):  
Tosiro Tsuzuku
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Irreducible Character ◽  
Symmetric Groups ◽  
Transitive Group ◽  
Permutation Groups ◽  
Irreducible Characters ◽  
Interesting Theorem

It is well known that a doubly transitive group has an irreducible character χ1 such that χ1(R) = α(R) − 1 for any element R of and a quadruply transitive group has irreducible characters χ3 and χ3 such that χ2(R) = where α(R) and β(R) are respectively the numbers of one cycles and two cycles contained in R. G. Frobenius was led to this fact in the connection with characters of the symmetric groups and he proved the following interesting theorem: if a permutation group of degree n is t-ply transitive, then any irreducible character of the symmetric group of degree n with dimension at most equal to is an irreducible character of .

scholarly journals A Character Condition for Quadruple Transitivity

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
S. Aldhafeeri ◽  
R. T. Curtis
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Irreducible Character

Let be a permutation group of degree viewed as a subgroup of the symmetric group . We show that if the irreducible character of corresponding to the partition of into subsets of sizes and 2, that is, to say the character often denoted by , remains irreducible when restricted to , then = 4, 5 or 9 and , A5, or PΣL2(8), respectively, or is 4-transitive.

Coprime Group Actions Fixing All Nonlinear Irreducible Characters

1989 ◽  
Vol 41 (1) ◽  
pp. 68-82 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Group Actions ◽  
Character Degrees ◽  
Prime Divisors ◽  
Irreducible Characters ◽  
Nilpotent Length ◽  
Derived Subgroup ◽  
Nonlinear Irreducible Character

The main result of this paper is the following:Theorem A. Let H and N be finite groups with coprime orders andsuppose that H acts nontrivially on N via automorphisms. Assume that Hfixes every nonlinear irreducible character of N. Then the derived subgroup ofN is nilpotent and so N is solvable of nilpotent length≦ 2.Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group Gfrom a knowledge of the setcd(G) = ﹛x(l)lx ∈ Irr(G) ﹜of irreducible character degrees of G.Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider π-separable groups (for some set π of primes) with the property that every element of cd(G) is either a 77-number or a π'-number.

Character correspondences in solvable groups with a self-normalizing sylow subgroup

2019 ◽  
Vol 19 (10) ◽  
pp. 2050190
Author(s):  
Carolina Vallejo Rodríguez
Keyword(s):  
Solvable Group ◽  
Irreducible Character ◽  
Sylow Subgroup ◽  
Solvable Groups ◽  
Character Formula ◽  
Finite Solvable Group ◽  
Irreducible Characters ◽  
Made In

Let [Formula: see text] be a finite solvable group and let [Formula: see text] for some prime [Formula: see text]. Whenever [Formula: see text] is odd, Isaacs described a correspondence between irreducible characters of degree not divisible by [Formula: see text] of [Formula: see text] and [Formula: see text]. This correspondence is natural in the sense that an algorithm is provided to compute it, and the result of the application of the algorithm does not depend on choices made. In the case where [Formula: see text], G. Navarro showed that every irreducible character [Formula: see text] of degree not divisible by [Formula: see text] has a unique linear constituent [Formula: see text] when restricted to [Formula: see text], and that the map [Formula: see text] defines a bijection. Navarro’s bijection is obviously natural in the sense described above. We show that these two correspondences are the same under the intersection of the hypotheses.

Semilinear preservers of the immanants in the set of the doubly stochastic matrices

2017 ◽  
Vol 32 ◽  
pp. 76-97
Author(s):  
M. Antonia Duffner ◽  
Rosario Fernandes
Keyword(s):  
Symmetric Group ◽  
Real Number ◽  
Irreducible Character ◽  
Complex Field ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices

Let $S_n$ denote the symmetric group of degree $n$ and $M_n$ denote the set of all $n$-by-$n$ matrices over the complex field, $\IC$. Let $\chi: S_n\rightarrow \IC$ be an irreducible character of degree greater than $1$ of $S_n$. The immanant $\dc: M_n \rightarrow \IC$ associated with $\chi$ is defined by $$ \dc(X) = \sum_{\sigma \in S_n} \chi(\sigma) \prod_{j=1}^n X_{j\sigma(j)} , \quad X = [X_{jk}] \in M_n. $$ Let $\Omega_n$ be the set of all $n$-by-$n$ doubly stochastic matrices, that is, matrices with nonnegative real entries and each row and column sum is one. We say that a map $T$ from $\Omega_n$ into $\Omega_n$ \begin{itemize} \item is semilinear if $T(\lambda S_1+(1-\lambda )S_2)=\lambda T(S_1)+(1-\lambda )T(S_2)$ for all $S_1,\ S_2\in \Omega_n$ and for all real number $\lambda$ such that $0\leq \lambda\leq 1$; \item preserves $d_{\chi }$ if $d_{\chi }(T(S))=d_{\chi }(S)$ for all $S\in\Omega_n$. \end{itemize} We characterize the semilinear surjective maps $T$ from $\Omega_n $ into $\Omega_n$ that preserve $\dc$, when the degree of $\chi$ is greater than one.

scholarly journals Blocks and Normal Subgroups of Finite Groups

1963 ◽  
Vol 22 ◽  
pp. 15-32 ◽  
Author(s):  
W. F. Reynolds
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Irreducible Character ◽  
Normal Subgroups ◽  
Irreducible Characters ◽  
Section 8 ◽  
A Finite Group

Let H be a normal subgroup of a finite group G, and let ζ be an (absolutely) irreducible character of H. In [7], Clifford studied the irreducible characters X of G whose restrictions to H contain ζ as a constituent. First he reduced this question to the same question in the so-called inertial subgroup S of ζ in G, and secondly he described the situation in S in terms of certain projective characters of S/H. In section 8 of [10], Mackey generalized these results to the situation where all the characters concerned are projective.

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.

SYMMETRY CLASSES OF POLYNOMIALS ASSOCIATED WITH THE DICYCLIC GROUP

2013 ◽  
Vol 06 (03) ◽  
pp. 1350033 ◽  
Author(s):  
Yousef Zamani ◽  
Esmaeil Babaei
Keyword(s):  
Symmetric Group ◽  
Quaternion Group ◽  
Irreducible Characters ◽  
Symmetry Classes ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

In this paper, we obtain the dimensions of symmetry classes of polynomials with respect to the irreducible characters of the dicyclic group as a subgroup of the full symmetric group. Then we discuss the existence of o-basis of these classes. In particular, the existence of o-basis of symmetry classes of polynomials with respect to the irreducible characters of the generalized quaternion group are concluded.

scholarly journals Murnaghan-Nakayama Rules for Characters of Iwahori-Hecke Algebras of the Complex Reflection Groups G(r, p, n)

1998 ◽  
Vol 50 (1) ◽  
pp. 167-192 ◽  
Author(s):  
Tom Halverson ◽  
Arun Ram
Keyword(s):  
Symmetric Group ◽  
Infinite Series ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Diagonal Matrix ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Irreducible Characters ◽  
Complex Reflection ◽  
Complex Reflection Groups

AbstractIwahori-Hecke algebras for the infinite series of complex reflection groups G(r, p, n) were constructed recently in the work of Ariki and Koike [AK], Broué andMalle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in whichwe derivedMurnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of G(r, p, n) given by Ariki and Koike [AK] and Ariki [Ari].

scholarly journals Stability of Kronecker Products of Irreducible Characters of the Symmetric Group

10.37236/1471 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Ernesto Vallejo
Keyword(s):  
Lower Bound ◽  
Symmetric Group ◽  
Kronecker Product ◽  
Kronecker Products ◽  
Irreducible Characters ◽  
Long Time ◽  
The Stability

F. Murnaghan observed a long time ago that the computation of the decompositon of the Kronecker product $\chi^{(n-a, \lambda_2, \dots )}\otimes \chi^{(n-b, \mu_2, \dots)}$ of two irreducible characters of the symmetric group into irreducibles depends only on $\overline\lambda=(\lambda_2,\dots )$ and $\overline\mu =(\mu_2,\dots )$, but not on $n$. In this note we prove a similar result: given three partitions $\lambda$, $\mu$, $\nu$ of $n$ we obtain a lower bound on $n$, depending on $\overline\lambda$, $\overline\mu$, $\overline\nu$, for the stability of the multiplicity $c(\lambda,\mu,\nu)$ of $\chi^\nu$ in $\chi^\lambda \otimes \chi^\mu$. Our proof is purely combinatorial. It uses a description of the $c(\lambda,\mu,\nu)$'s in terms of signed special rim hook tabloids and Littlewood-Richardson multitableaux.

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