On Self-Mullineux and Self-Conjugate Partitions

Ana Bernal

doi:10.37236/9283

On Self-Mullineux and Self-Conjugate Partitions

The Electronic Journal of Combinatorics ◽

10.37236/9283 ◽

2021 ◽

Vol 28 (1) ◽

Author(s):

Ana Bernal

Keyword(s):

Symmetric Group ◽

Fixed Points ◽

Alternating Group ◽

Modular Representations ◽

Distinguished Subset

The Mullineux involution is a relevant map that appears in the study of the modular representations of the symmetric group and the alternating group. The fixed points of this map are certain partitions of particular interest. It is known that the cardinality of the set of these self-Mullineux partitions is equal to the cardinality of a distinguished subset of self-conjugate partitions. In this work, we give an explicit bijection between the two families of partitions in terms of the Mullineux symbol.

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On the Modular Representations of the Symmetric Group

Canadian Journal of Mathematics ◽

10.4153/cjm-1954-052-9 ◽

1954 ◽

Vol 6 ◽

pp. 486-497 ◽

Cited By ~ 1

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 ).

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Projective modular representations of finite groups II

Glasgow Mathematical Journal ◽

10.1017/s0017089500004511 ◽

1981 ◽

Vol 22 (1) ◽

pp. 89-99 ◽

Cited By ~ 8

Author(s):

J. F. Humphreys

Keyword(s):

Finite Group ◽

Finite Groups ◽

Theoretical Background ◽

Alternating Group ◽

Modular Representations ◽

Future Paper ◽

Representations Of Finite Groups ◽

Group A ◽

Mathieu Groups

In this paper, which is a continuation of [4], the necessary theoretical background is given to enable the calculation of the irreducible Brauer projective characters of a given finite group to be carried out. As an example, this calculation is done for the alternating group A (7) in §3. In a future paper the calculations for the Mathieu groups will be presented.

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A Class of Trinomials with Galois Group Sn

Algebra Colloquium ◽

10.1142/s1005386712000764 ◽

2012 ◽

Vol 19 (spec01) ◽

pp. 905-911 ◽

Cited By ~ 1

Author(s):

Anuj Bishnoi ◽

Sudesh K. Khanduja

Keyword(s):

Positive Integer ◽

Symmetric Group ◽

Galois Group ◽

Rational Numbers ◽

Alternating Group ◽

Polynomial Formula

A well known result of Schur states that if n is a positive integer and a0, a1,…,an are arbitrary integers with a0an coprime to n!, then the polynomial [Formula: see text] is irreducible over the field ℚ of rational numbers. In case each ai = 1, it is known that the Galois group of fn(x) over ℚ contains An, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials fn(x) which leads to the construction of trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters.

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Modular Representations of Sn

Canadian Journal of Mathematics ◽

10.4153/cjm-1964-019-9 ◽

1964 ◽

Vol 16 ◽

pp. 191-203 ◽

Cited By ~ 3

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

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Irreducible Modular Representations of the Reflection GroupG(m,1,n)

Journal of Mathematics ◽

10.1155/2015/808520 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9

Author(s):

José O. Araujo ◽

Tim Bratten ◽

Cesar L. Maiarú

Keyword(s):

Symmetric Group ◽

Weyl Group ◽

Reflection Group ◽

Geometric Construction ◽

Weyl Groups ◽

Modular Representations ◽

Complex Reflection ◽

Complex Reflection Group ◽

One Year ◽

Combinatorial Methods

In an article published in 1980, Farahat and Peel realized the irreducible modular representations of the symmetric group. One year later, Al-Aamily, Morris, and Peel constructed the irreducible modular representations for a Weyl group of typeBn. In both cases, combinatorial methods were used. Almost twenty years later, using a geometric construction based on the ideas of Macdonald, first Aguado and Araujo and then Araujo, Bigeón, and Gamondi also realized the irreducible modular representations for the Weyl groups of typesAnandBn. In this paper, we extend the geometric construction based on the ideas of Macdonald to realize the irreducible modular representations of the complex reflection group of typeG(m,1,n).

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Integral Cayley Graphs over Finite Groups

Algebra Colloquium ◽

10.1142/s1005386720000115 ◽

2020 ◽

Vol 27 (01) ◽

pp. 131-136

Author(s):

Elena V. Konstantinova ◽

Daria Lytkina

Keyword(s):

Finite Group ◽

Symmetric Group ◽

Cayley Graph ◽

Cyclic Group ◽

Finite Groups ◽

Cayley Graphs ◽

Alternating Group ◽

Generating Set ◽

A Finite Group

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

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Finite Groups Which Factor as Product of an Alternating Group and a Symmetric Group

Communications in Algebra ◽

10.1081/agb-120027919 ◽

2004 ◽

Vol 32 (2) ◽

pp. 637-647 ◽

Cited By ~ 2

Author(s):

M. R. Darafsheh

Keyword(s):

Symmetric Group ◽

Finite Groups ◽

Alternating Group

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Galois groups of Taylor polynomials of some elementary functions

International Journal of Number Theory ◽

10.1142/s1793042119500623 ◽

2019 ◽

Vol 15 (06) ◽

pp. 1127-1141

Author(s):

Khosro Monsef Shokri ◽

Jafar Shaffaf ◽

Reza Taleb

Keyword(s):

Symmetric Group ◽

Galois Groups ◽

Alternating Group ◽

Integer Number ◽

Elementary Functions ◽

Type Formula ◽

Number Formula ◽

Taylor Polynomials ◽

Index 2

Motivated by Schur’s result on computing the Galois groups of the exponential Taylor polynomials, this paper aims to compute the Galois groups of the Taylor polynomials of the elementary functions [Formula: see text] and [Formula: see text]. We first show that the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] are as large as possible, namely, [Formula: see text] (full symmetric group) or [Formula: see text] (alternating group), depending on the residue of the integer number [Formula: see text] modulo [Formula: see text]. We then compute the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] and show that these Galois groups essentially coincide with the Coexter groups of type [Formula: see text] (or an index 2 subgroup of the corresponding Coexter group).

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